MYP 4 & 5 · Maths
INDICES
60 questions across 5 sub-topics
Use the Sub-Topic filter above to focus on one.
Index notation
Index laws
Exponential equations
Scientific notation (Standard form)
Rational (fractional) indices
Index notation 12 questions
QUESTION 1
4 marks
Criterion A
Easy
Index notation is used to write repeated multiplication in a shorter form.
a.
Write $6 \times 6 \times 6 \times 6$ as a single power.
[1] b.
Write $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ as a single power, then evaluate it.
[2] c.
Evaluate $5^3$.
[1] Show complete worked solution
(a)
$6^4$
(b)
$2^7 = 128$
(c)
$125$
QUESTION 2
5 marks
Criterion B
Medium
Study the pattern of powers of $3$ below.$$3^4=81,\quad 3^3=27,\quad 3^2=9,\quad 3^1=3,\quad 3^0=\ ?$$ Each row is found by dividing the previous value by $3$.
a.
Determine the value of $3^0$, by continuing the pattern.
[1] b.
State a general rule for $a^0$, where $a \neq 0$.
[1] c.
Continue the pattern one more step to determine the value of $3^{-1}$.
[2] d.
Hence write down the value of $3^{-2}$.
[1] Show complete worked solution
(a)
$9 \div 3 = 3$, then $3 \div 3 = 1$, so $3^0 = 1$.
(b)
$a^0 = 1$
(c)
$1 \div 3 = \dfrac{1}{3}$, so $3^{-1} = \dfrac{1}{3}$.
(d)
$\dfrac{1}{3} \div 3 = \dfrac{1}{9}$
QUESTION 3
3 marks
Criterion A
Easy
a.
In the expression $7^5$, state the base and the exponent (index).
[1] b.
Write $x \times x \times x \times x$ as a power of $x$.
[1] c.
Write $2 \times 2 \times 2 \times 5 \times 5$ using index notation.
[1] Show complete worked solution
(a)
Base $=7$, exponent (index) $=5$.
(b)
$x^4$
(c)
$2^3 \times 5^2$
QUESTION 4
4 marks
Criterion B
Medium
a.
Evaluate $2^{-3}$.
[1] b.
Evaluate $5^{-2}$.
[1] c.
Evaluate $4^{-1}$.
[1] d.
Determine the value of $10^{-3}$, giving your answer as a decimal.
[1] Show complete worked solution
(a)
$\dfrac{1}{8}$
(b)
$\dfrac{1}{25}$
(c)
$\dfrac{1}{4}$
(d)
$0.001$
QUESTION 5
6 marks
Criterion C
Medium
Consider the powers $(-2)^2$, $(-2)^3$, $(-2)^4$ and $(-2)^5$.
a.
Evaluate $(-2)^2$ and $(-2)^3$.
[2] b.
Evaluate $(-2)^4$ and $(-2)^5$.
[2] c.
Explain the rule for the sign of $(-a)^n$, distinguishing between even and odd values of $n$.
[2] Show complete worked solution
(a)
$(-2)^2 = 4$, $(-2)^3 = -8$
(b)
$(-2)^4 = 16$, $(-2)^5 = -32$
(c)
When $n$ is even, $(-a)^n$ is positive (the negative signs cancel in pairs). When $n$ is odd, $(-a)^n$ is negative (one negative sign is left over after pairing).
QUESTION 6
4 marks
Criterion C
Medium
a.
Evaluate $(-3)^4$.
[1] b.
Evaluate $-3^4$ (i.e. the negative of $3$ raised to the power $4$).
[1] c.
Explain why these two expressions give different answers even though they look similar.
[2] Show complete worked solution
(a)
$81$
(b)
$-81$
(c)
In $(-3)^4$ the base is $-3$, so the whole negative number is raised to the power. In $-3^4$ only $3$ is raised to the power, and the negative sign is applied afterwards (exponentiation is carried out before the unary minus).
QUESTION 7
6 marks
Criterion A
Easy
a.
Evaluate $2^3 + 3^2$.
[2] b.
Evaluate $5^2 - 2^4$.
[2] c.
Evaluate $4^0 \times 6^2$.
[2] Show complete worked solution
(a)
$8 + 9 = 17$
(b)
$25 - 16 = 9$
(c)
$1 \times 36 = 36$
QUESTION 8
6 marks
Criterion D
Medium
A single bacterium divides into $2$ bacteria every hour. After $t$ hours, the number of bacteria in the colony is given by $2^t$.
a.
Write down the number of bacteria after $0$ hours.
[1] b.
Determine the number of bacteria after $5$ hours.
[2] c.
Determine, by evaluating successive powers of $2$, after how many whole hours the population will first exceed $1000$.
[3] Show complete worked solution
(a)
$2^0 = 1$
(b)
$2^5 = 32$
(c)
$2^9 = 512$ (not yet exceeded), $2^{10} = 1024$ (exceeded). The population first exceeds $1000$ after $10$ hours.
QUESTION 9
6 marks
Criterion C
Hard
a.
Evaluate $2^{10}$.
[1] b.
Evaluate $10^2$.
[1] c.
Hence determine which is greater, $2^{10}$ or $10^2$, and by how much.
[2] d.
Explain, without calculating $2^{20}$ exactly, why $2^{20}$ must be far greater than $20^2$.
[2] Show complete worked solution
(a)
$1024$
(b)
$100$
(c)
$2^{10}$ is greater, by $1024 - 100 = 924$.
(d)
As the exponent increases by $1$, a power such as $2^n$ is multiplied by the base each time, so it grows faster and faster. A squared term like $20^2$ only ever involves multiplying two factors of $20$ together, no matter how the exponent notation looks, so it cannot keep pace with a quantity that keeps doubling. (In fact $2^{20} = 1\,048\,576$, while $20^2 = 400$.)
QUESTION 10
6 marks
Criterion A
Medium
a.
Evaluate $6^2$ and $6^3$.
[2] b.
Determine, by trial, the value of $n$ such that $n^2 = 169$.
[2] c.
Determine the value of $n$ such that $n^3 = 125$.
[2] Show complete worked solution
(a)
$6^2 = 36$, $6^3 = 216$
(b)
$n = 13$, since $13^2 = 169$.
(c)
$n = 5$, since $5^3 = 125$.
QUESTION 11
4 marks
Criterion B
Medium
The table below shows values of $2^n$ for consecutive values of $n$.$$n=1,2,3,4,5 \quad \rightarrow \quad 2^n = 2,4,8,16,32$$
a.
State the value of $2^6$, by continuing the pattern (doubling each time).
[1] b.
Describe, in words, the pattern used to generate each successive value of $2^n$ from the previous one.
[1] c.
Using the pattern, determine the value of $n$ for which $2^n = 256$.
[2] Show complete worked solution
(a)
$64$
(b)
Each value is double the previous value, since increasing $n$ by $1$ means multiplying by one more factor of $2$.
(c)
Continuing: $2^6=64$, $2^7=128$, $2^8=256$. So $n = 8$.
QUESTION 12
6 marks
Criterion C
Hard
a.
Evaluate $5^0 + 5^{-1} + 5^1$, giving your answer as a fraction.
[3] b.
Evaluate $(-1)^{100}$ and $(-1)^{101}$.
[2] c.
Hence evaluate $(-1)^{100} + (-1)^{101}$.
[1] Show complete worked solution
(a)
$1 + \dfrac{1}{5} + 5 = \dfrac{5}{5}+\dfrac{1}{5}+\dfrac{25}{5} = \dfrac{31}{5}$
(b)
$(-1)^{100} = 1$ (even exponent), $(-1)^{101} = -1$ (odd exponent)
(c)
$1 + (-1) = 0$
Index laws 12 questions
QUESTION 1
4 marks
Criterion A
Easy
a.
Simplify $x^3 \times x^5$, using an index law.
[1] b.
Simplify $2^4 \times 2^3$, then evaluate the result.
[2] c.
Simplify $y \times y^4 \times y^2$.
[1] Show complete worked solution
(a)
$x^8$
(b)
$2^7 = 128$
(c)
$y^7$
QUESTION 2
4 marks
Criterion A
Easy
a.
Simplify $x^7 \div x^2$.
[1] b.
Simplify $\dfrac{3^9}{3^4}$, then evaluate the result.
[2] c.
Simplify $\dfrac{a^6}{a^6}$.
[1] Show complete worked solution
(a)
$x^5$
(b)
$3^5 = 243$
(c)
$a^0 = 1$
QUESTION 3
5 marks
Criterion B
Medium
a.
Simplify $(x^3)^4$.
[1] b.
Simplify $(2^2)^3$, then evaluate the result.
[2] c.
Simplify $(y^{-2})^3$, giving your answer using positive indices.
[2] Show complete worked solution
(a)
$x^{12}$
(b)
$2^6 = 64$
(c)
$y^{-6} = \dfrac{1}{y^6}$
QUESTION 4
5 marks
Criterion B
Medium
a.
Simplify $(3x)^2$.
[1] b.
Simplify $(2xy)^3$.
[2] c.
Simplify $\left(\dfrac{x}{2}\right)^3$.
[2] Show complete worked solution
(a)
$9x^2$
(b)
$8x^3y^3$
(c)
$\dfrac{x^3}{8}$
QUESTION 5
6 marks
Criterion C
Medium
Consider the expression $(4x^3)(3x^5)$.
a.
Simplify the expression fully.
[2] b.
Hence evaluate the expression when $x = 2$.
[2] c.
A second expression is $(5x^2)(2x^4)$. Simplify this expression.
[2] Show complete worked solution
(a)
$(4x^3)(3x^5) = 12x^8$
(b)
$12 \times 2^8 = 12 \times 256 = 3072$
(c)
$10x^6$
QUESTION 6
7 marks
Criterion C
Medium
a.
Simplify $\dfrac{12x^7}{4x^3}$.
[2] b.
Simplify $\dfrac{18x^5y^3}{6x^2y}$.
[3] c.
Simplify $\dfrac{20a^4b^6}{5a^4b^2}$.
[2] Show complete worked solution
(a)
$3x^4$
(b)
$3x^3y^2$
(c)
$4b^4$
QUESTION 7
5 marks
Criterion C
Hard
Simplify $\dfrac{(2x^2)^3 \times x^4}{4x^5}$, showing each step of your working.
a.
Simplify $(2x^2)^3$.
[2] b.
Hence simplify $(2x^2)^3 \times x^4$.
[1] c.
Hence simplify $\dfrac{(2x^2)^3 \times x^4}{4x^5}$ fully.
[2] Show complete worked solution
(a)
$2^3 (x^2)^3 = 8x^6$
(b)
$8x^6 \times x^4 = 8x^{10}$
(c)
$\dfrac{8x^{10}}{4x^5} = 2x^5$
QUESTION 8
6 marks
Criterion D
Hard
Simplify $\dfrac{x^3y^2}{x^5y^{-1}}$, giving your answer using positive indices only.
a.
Simplify the $x$ terms, $\dfrac{x^3}{x^5}$, giving your answer using a positive index.
[2] b.
Simplify the $y$ terms, $\dfrac{y^2}{y^{-1}}$.
[2] c.
Hence write $\dfrac{x^3y^2}{x^5y^{-1}}$ in simplest form, using positive indices only.
[2] Show complete worked solution
(a)
$x^{3-5} = x^{-2} = \dfrac{1}{x^2}$
(b)
$y^{2-(-1)} = y^3$
(c)
$\dfrac{y^3}{x^2}$
QUESTION 9
3 marks
Criterion A
Medium
a.
Simplify $\left(\dfrac{2}{3}\right)^3$ as a fraction.
[1] b.
Simplify $\left(\dfrac{5}{2}\right)^2$ as a fraction.
[1] c.
Evaluate $\left(\dfrac{5}{2}\right)^2$ as a decimal.
[1] Show complete worked solution
(a)
$\dfrac{8}{27}$
(b)
$\dfrac{25}{4}$
(c)
$6.25$
QUESTION 10
5 marks
Criterion B
Hard
Consider $a^3 \times a^2$.
a.
Write $a^3 \times a^2$ as a single product of repeated factors of $a$ (expand both powers and combine).
[1] b.
Hence show that $a^3 \times a^2 = a^5$.
[1] c.
Using the same reasoning with general exponents $m$ and $n$, state the general index law for $a^m \times a^n$.
[1] d.
Hence, or otherwise, simplify $b^4 \times b^7 \times b$.
[2] Show complete worked solution
(a)
$a \times a \times a \times a \times a$ ($5$ factors of $a$)
(b)
$5$ factors of $a$ multiplied together is $a^5$.
(c)
$a^m \times a^n = a^{m+n}$
(d)
$b^{4+7+1} = b^{12}$
QUESTION 11
5 marks
Criterion C
Medium
a.
Simplify $x^5 \times x^{-3}$.
[1] b.
Simplify $x^{-2} \times x^{-4}$, using a positive index in your answer.
[2] c.
Simplify $\dfrac{x^2}{x^{-3}}$.
[2] Show complete worked solution
(a)
$x^2$
(b)
$x^{-6} = \dfrac{1}{x^6}$
(c)
$x^5$
QUESTION 12
7 marks
Criterion D
Hard
A cube has side length $s$ cm. A second, larger cube has side length $3s$ cm.
a.
Write down an expression, in terms of $s$, for the volume of the first cube.
[1] b.
Write an expression, in terms of $s$, for the volume of the second cube, giving your answer in simplified index form.
[2] c.
Hence determine how many times greater the volume of the second cube is compared to the first.
[1] d.
The surface area of a cube with side $s$ is $6s^2$. Determine, in simplified form, the surface area of the second cube in terms of $s$, and state how many times greater it is than the first cube's surface area.
[3] Show complete worked solution
(a)
$s^3$
(b)
$(3s)^3 = 27s^3$
(c)
$\dfrac{27s^3}{s^3} = 27$ times greater.
(d)
$6(3s)^2 = 6 \times 9s^2 = 54s^2$. Since $\dfrac{54s^2}{6s^2} = 9$, the surface area is $9$ times greater.
Exponential equations 12 questions
QUESTION 1
3 marks
Criterion A
Easy
a.
Solve $2^x = 32$ for $x$.
[1] b.
Solve $5^x = 125$ for $x$.
[1] c.
Solve $3^x = 1$ for $x$.
[1] Show complete worked solution
(a)
$2^x = 2^5 \Rightarrow x = 5$
(b)
$5^x = 5^3 \Rightarrow x = 3$
(c)
$3^x = 3^0 \Rightarrow x = 0$
QUESTION 2
4 marks
Criterion A
Easy
a.
Solve $2^x = \dfrac{1}{8}$ for $x$.
[2] b.
Solve $10^x = 0.01$ for $x$.
[2] Show complete worked solution
(a)
$2^x = 2^{-3} \Rightarrow x = -3$
(b)
$10^x = 10^{-2} \Rightarrow x = -2$
QUESTION 3
6 marks
Criterion B
Medium
a.
Solve $2^{x+1} = 16$ for $x$.
[2] b.
Solve $3^{2x} = 81$ for $x$.
[2] c.
Solve $5^{x-2} = 25$ for $x$.
[2] Show complete worked solution
(a)
$2^{x+1} = 2^4 \Rightarrow x+1=4 \Rightarrow x = 3$
(b)
$3^{2x} = 3^4 \Rightarrow 2x = 4 \Rightarrow x = 2$
(c)
$5^{x-2} = 5^2 \Rightarrow x-2=2 \Rightarrow x = 4$
QUESTION 4
5 marks
Criterion C
Medium
a.
Express $8$ as a power of $2$.
[1] b.
Hence solve $8^x = 32$ for $x$.
[2] c.
Solve $4^x = 128$ for $x$, giving your answer as a fraction.
[2] Show complete worked solution
(a)
$8 = 2^3$
(b)
$(2^3)^x = 2^5 \Rightarrow 3x = 5 \Rightarrow x = \dfrac{5}{3}$
(c)
$(2^2)^x = 2^7 \Rightarrow 2x = 7 \Rightarrow x = \dfrac{7}{2}$
QUESTION 5
6 marks
Criterion C
Hard
a.
Solve $2^{3x-1} = 2^{x+5}$ for $x$.
[3] b.
Solve $3^{2x+1} = 3^{x+4}$ for $x$.
[3] Show complete worked solution
(a)
$3x-1 = x+5 \Rightarrow 2x = 6 \Rightarrow x = 3$
(b)
$2x+1 = x+4 \Rightarrow x = 3$
QUESTION 6
6 marks
Criterion D
Hard
A population of bacteria doubles every hour. The population, $P$, after $t$ hours is modelled by $P = 500 \times 2^t$.
a.
Determine the initial population (at $t=0$).
[1] b.
Determine the population after $4$ hours.
[2] c.
Determine the value of $t$ for which the population first equals $16\,000$.
[3] Show complete worked solution
(a)
$500 \times 2^0 = 500$
(b)
$500 \times 2^4 = 500 \times 16 = 8000$
(c)
$500 \times 2^t = 16\,000 \Rightarrow 2^t = 32 = 2^5 \Rightarrow t = 5$
QUESTION 7
6 marks
Criterion B
Medium
a.
Solve $\left(\dfrac{1}{2}\right)^x = 16$ for $x$.
[2] b.
Solve $\left(\dfrac{1}{3}\right)^x = \dfrac{1}{81}$ for $x$.
[2] c.
Solve $\left(\dfrac{1}{4}\right)^x = 64$ for $x$.
[2] Show complete worked solution
(a)
$(2^{-1})^x = 2^4 \Rightarrow -x = 4 \Rightarrow x = -4$
(b)
$(3^{-1})^x = 3^{-4} \Rightarrow -x = -4 \Rightarrow x = 4$
(c)
$(4^{-1})^x = 4^3 \Rightarrow -x = 3 \Rightarrow x = -3$
QUESTION 8
6 marks
Criterion C
Medium
a.
Simplify $2^x \times 2^3$ to a single power of $2$.
[1] b.
Hence solve $2^x \times 2^3 = 64$ for $x$.
[2] c.
Solve $3 \times 3^x = 243$ for $x$ (Hint: write $3$ as $3^1$ first).
[3] Show complete worked solution
(a)
$2^{x+3}$
(b)
$2^{x+3} = 2^6 \Rightarrow x+3=6 \Rightarrow x = 3$
(c)
$3^{1+x} = 3^5 \Rightarrow 1+x = 5 \Rightarrow x = 4$
QUESTION 9
6 marks
Criterion D
Hard
A radioactive sample has mass $M$ grams after $t$ years, given by $M = 800 \times \left(\dfrac{1}{2}\right)^t$.
a.
Determine the initial mass of the sample.
[1] b.
Determine the mass remaining after $3$ years.
[2] c.
Determine the value of $t$ for which only $25$ grams remain.
[3] Show complete worked solution
(a)
$800$ g
(b)
$800 \times \left(\dfrac{1}{2}\right)^3 = 800 \times \dfrac{1}{8} = 100$ g
(c)
$800 \times \left(\dfrac{1}{2}\right)^t = 25 \Rightarrow \left(\dfrac{1}{2}\right)^t = \dfrac{1}{32} = \left(\dfrac{1}{2}\right)^5 \Rightarrow t = 5$
QUESTION 10
5 marks
Criterion A
Medium
Three equations and three proposed solutions are shown: (i) $2^x=64,\ x=5$; (ii) $5^x=625,\ x=4$; (iii) $3^x=1,\ x=1$.
a.
Determine, with working, whether the proposed solution to equation (i) is correct.
[1] b.
Determine, with working, whether the proposed solution to equation (ii) is correct.
[1] c.
Determine, with working, whether the proposed solution to equation (iii) is correct.
[1] d.
State the correct solution for each equation you found to be incorrect.
[2] Show complete worked solution
(a)
$2^5 = 32 \neq 64$, so incorrect.
(b)
$5^4 = 625$, so correct.
(c)
$3^1 = 3 \neq 1$, so incorrect.
(d)
(i) $2^x = 64 = 2^6 \Rightarrow x = 6$. (iii) $3^x = 1 = 3^0 \Rightarrow x = 0$.
QUESTION 11
4 marks
Criterion B
Medium
a.
Evaluate $2^4$ and $2^5$.
[2] b.
Hence determine between which two consecutive integers $x$ must lie, if $2^x = 20$, explaining your reasoning.
[2] Show complete worked solution
(a)
$2^4 = 16$, $2^5 = 32$
(b)
Since $16 < 20 < 32$, i.e. $2^4 < 20 < 2^5$, the value of $x$ must lie between $4$ and $5$.
QUESTION 12
7 marks
Criterion D
Hard
A company's number of app downloads triples every month. In month $0$ there were $9$ downloads.
a.
Write down an expression, in terms of $t$, for the number of downloads after $t$ months.
[2] b.
Determine the number of downloads after $3$ months.
[2] c.
Determine the value of $t$ for which the number of downloads first equals $2187$.
[3] Show complete worked solution
(a)
$9 \times 3^t$
(b)
$9 \times 3^3 = 9 \times 27 = 243$
(c)
$9 \times 3^t = 2187 \Rightarrow 3^t = 243 = 3^5 \Rightarrow t = 5$
Scientific notation (Standard form) 12 questions
QUESTION 1
3 marks
Criterion A
Easy
a.
Write $47\,000$ in standard form.
[1] b.
Write $6\,300\,000$ in standard form.
[1] c.
Write $520$ in standard form.
[1] Show complete worked solution
(a)
$4.7 \times 10^4$
(b)
$6.3 \times 10^6$
(c)
$5.2 \times 10^2$
QUESTION 2
3 marks
Criterion A
Easy
a.
Write $0.00034$ in standard form.
[1] b.
Write $0.0052$ in standard form.
[1] c.
Write $0.000009$ in standard form.
[1] Show complete worked solution
(a)
$3.4 \times 10^{-4}$
(b)
$5.2 \times 10^{-3}$
(c)
$9 \times 10^{-6}$
QUESTION 3
3 marks
Criterion A
Easy
a.
Write $3.6 \times 10^5$ as an ordinary number.
[1] b.
Write $7.1 \times 10^{-3}$ as an ordinary number.
[1] c.
Write $2 \times 10^7$ as an ordinary number.
[1] Show complete worked solution
(a)
$360\,000$
(b)
$0.0071$
(c)
$20\,000\,000$
QUESTION 4
5 marks
Criterion B
Medium
The expression $42 \times 10^5$ is not written in correct standard form.
a.
Explain why $42 \times 10^5$ is not in correct standard form.
[1] b.
Rewrite $42 \times 10^5$ in correct standard form.
[2] c.
Rewrite $0.36 \times 10^{-2}$ in correct standard form.
[2] Show complete worked solution
(a)
In standard form $a \times 10^n$, the coefficient $a$ must satisfy $1 \le a < 10$ — here $a = 42$, which is too large.
(b)
$4.2 \times 10^6$
(c)
$3.6 \times 10^{-3}$
QUESTION 5
7 marks
Criterion C
Medium
a.
Simplify $(2 \times 10^3) \times (4 \times 10^5)$, giving your answer in standard form.
[2] b.
Simplify $(3 \times 10^{-2}) \times (2 \times 10^6)$, giving your answer in standard form.
[2] c.
Simplify $(5 \times 10^4) \times (6 \times 10^3)$, giving your answer in standard form.
[3] Show complete worked solution
(a)
$8 \times 10^8$
(b)
$6 \times 10^4$
(c)
$30 \times 10^7 = 3 \times 10^8$ (the coefficient must be renormalised since $30 \geq 10$)
QUESTION 6
7 marks
Criterion C
Medium
a.
Simplify $(8 \times 10^7) \div (2 \times 10^3)$, giving your answer in standard form.
[2] b.
Simplify $(9 \times 10^{-2}) \div (3 \times 10^4)$, giving your answer in standard form.
[2] c.
Simplify $(6 \times 10^5) \div (8 \times 10^2)$, giving your answer in standard form.
[3] Show complete worked solution
(a)
$4 \times 10^4$
(b)
$3 \times 10^{-6}$
(c)
$0.75 \times 10^3 = 7.5 \times 10^2$ (the coefficient must be renormalised since $0.75 < 1$)
QUESTION 7
7 marks
Criterion D
Hard
The distance from Earth to the Sun is approximately $1.5 \times 10^8$ km. The distance from Earth to the Moon is approximately $3.84 \times 10^5$ km.
a.
Write the Earth–Sun distance as an ordinary number.
[1] b.
Determine how many times greater the Earth–Sun distance is than the Earth–Moon distance, giving your answer to $3$ significant figures.
[3] c.
A spacecraft travels at $3 \times 10^4$ km/h. Determine, to the nearest hour, how long it would take to travel from the Earth to the Moon.
[3] Show complete worked solution
(a)
$150\,000\,000$ km
(b)
$\dfrac{1.5 \times 10^8}{3.84 \times 10^5} = 390.625 \approx 391$ (3 s.f.)
(c)
$\dfrac{3.84 \times 10^5}{3 \times 10^4} = 12.8$ hours $\approx 13$ hours
QUESTION 8
7 marks
Criterion D
Hard
A red blood cell has a diameter of approximately $7 \times 10^{-6}$ m. A white blood cell has a diameter of approximately $1.2 \times 10^{-5}$ m.
a.
Write both diameters as ordinary decimal numbers (in metres).
[2] b.
Determine the sum of the two diameters, giving your answer in standard form.
[3] c.
Determine how many times greater the white blood cell's diameter is compared to the red blood cell's, to $2$ significant figures.
[2] Show complete worked solution
(a)
Red blood cell: $0.000007$ m. White blood cell: $0.000012$ m.
(b)
$0.000007 + 0.000012 = 0.000019 = 1.9 \times 10^{-5}$ m
(c)
$\dfrac{1.2 \times 10^{-5}}{7 \times 10^{-6}} = 1.714\ldots \approx 1.7$ (2 s.f.)
QUESTION 9
4 marks
Criterion A
Medium
Consider the four numbers shown in the table.
| Label | Value |
|---|---|
| A | $3.2 \times 10^4$ |
| B | $5.6 \times 10^3$ |
| C | $2.9 \times 10^5$ |
| D | $8.1 \times 10^4$ |
a.
Write all four numbers as ordinary numbers.
[2] b.
Hence arrange the four numbers in ascending order.
[2] Show complete worked solution
(a)
A $=32\,000$, B $=5600$, C $=290\,000$, D $=81\,000$
(b)
B $<$ A $<$ D $<$ C, i.e. $5.6\times10^3 < 3.2\times10^4 < 8.1\times10^4 < 2.9\times10^5$
QUESTION 10
6 marks
Criterion B
Medium
The population of a country is approximately $6.4 \times 10^7$. Each person uses on average $2.5 \times 10^2$ litres of water per day.
a.
Determine the total volume of water used by the population in one day, giving your answer in standard form.
[3] b.
Determine the total volume of water used in one year ($365$ days), giving your answer in standard form to $3$ significant figures.
[3] Show complete worked solution
(a)
$6.4 \times 10^7 \times 2.5 \times 10^2 = 16 \times 10^9 = 1.6 \times 10^{10}$ litres
(b)
$1.6 \times 10^{10} \times 365 = 5.84 \times 10^{12}$ litres
QUESTION 11
7 marks
Criterion C
Hard
a.
Simplify $\dfrac{4 \times 10^{-3}}{2 \times 10^{-7}}$, giving your answer in standard form.
[3] b.
Simplify $(2 \times 10^{-4})^2$, giving your answer in standard form.
[2] c.
Simplify $(5 \times 10^3) + (2 \times 10^2)$, giving your answer in standard form.
[2] Show complete worked solution
(a)
$2 \times 10^{4}$
(b)
$4 \times 10^{-8}$
(c)
$5000 + 200 = 5200 = 5.2 \times 10^3$
QUESTION 12
4 marks
Criterion A
Medium
Determine whether each statement is true or false, giving a reason.
a.
“$5.2 \times 10^{-3}$ is a larger number than $5.2 \times 10^{3}$.”
[2] b.
“$3 \times 10^4 = 3000$.”
[1] c.
“For a number in standard form $a \times 10^n$, the coefficient $a$ must satisfy $1 \le a < 10$.”
[1] Show complete worked solution
(a)
False. $5.2 \times 10^{-3} = 0.0052$ while $5.2 \times 10^3 = 5200$, so the first is far smaller.
(b)
False. $3 \times 10^4 = 30\,000$.
(c)
True — this is the definition of standard form.
Rational (fractional) indices 12 questions
QUESTION 1
3 marks
Criterion A
Easy
a.
Evaluate $8^{1/3}$.
[1] b.
Evaluate $25^{1/2}$.
[1] c.
Evaluate $16^{1/4}$.
[1] Show complete worked solution
(a)
$2$
(b)
$5$
(c)
$2$
QUESTION 2
6 marks
Criterion A
Easy
a.
Write $49^{1/2}$ using root (radical) notation, then evaluate it.
[2] b.
Write $\sqrt[3]{27}$ using fractional index notation, then evaluate it.
[2] c.
Write $\sqrt[5]{32}$ using fractional index notation, then evaluate it.
[2] Show complete worked solution
(a)
$\sqrt{49} = 7$
(b)
$27^{1/3} = 3$
(c)
$32^{1/5} = 2$
QUESTION 3
6 marks
Criterion B
Medium
The expression $a^{m/n}$ can be evaluated as $\left(a^{1/n}\right)^m$ — first take the $n$th root of $a$, then raise the result to the power $m$.
a.
Using this method, evaluate $27^{2/3}$.
[2] b.
Evaluate $16^{3/4}$.
[2] c.
Evaluate $4^{3/2}$.
[2] Show complete worked solution
(a)
$\left(27^{1/3}\right)^2 = 3^2 = 9$
(b)
$\left(16^{1/4}\right)^3 = 2^3 = 8$
(c)
$\left(4^{1/2}\right)^3 = 2^3 = 8$
QUESTION 4
7 marks
Criterion C
Medium
a.
Evaluate $4^{-1/2}$.
[2] b.
Evaluate $8^{-2/3}$.
[2] c.
Evaluate $9^{-3/2}$.
[3] Show complete worked solution
(a)
$\left(4^{1/2}\right)^{-1} = 2^{-1} = \dfrac{1}{2}$
(b)
$\left(8^{1/3}\right)^{-2} = 2^{-2} = \dfrac{1}{4}$
(c)
$\left(9^{1/2}\right)^{-3} = 3^{-3} = \dfrac{1}{27}$
QUESTION 5
4 marks
Criterion B
Medium
Consider $x^{1/2} \times x^{1/2}$.
a.
Using the multiplication index law, simplify $x^{1/2} \times x^{1/2}$ to a single power of $x$.
[1] b.
Since $x^{1/2} \times x^{1/2} = x$, explain what this tells you about the meaning of $x^{1/2}$.
[1] c.
Using the same reasoning with $x^{1/3} \times x^{1/3} \times x^{1/3}$, determine what $x^{1/3}$ represents.
[2] Show complete worked solution
(a)
$x^{1/2+1/2} = x^1 = x$
(b)
$x^{1/2}$ is a number which, when multiplied by itself, gives $x$ — that is exactly the definition of the square root, so $x^{1/2} = \sqrt{x}$.
(c)
$x^{1/3} \times x^{1/3} \times x^{1/3} = x^{1/3+1/3+1/3} = x^1 = x$, so three copies of $x^{1/3}$ multiply to give $x$ — meaning $x^{1/3}$ is the cube root of $x$, $\sqrt[3]{x}$.
QUESTION 6
7 marks
Criterion C
Hard
a.
Simplify $x^{1/2} \times x^{3/2}$.
[1] b.
Simplify $\dfrac{x^{5/3}}{x^{2/3}}$.
[2] c.
Simplify $\left(x^{2/3}\right)^3$.
[2] d.
Simplify $\left(x^{1/2}y^2\right)^4$.
[2] Show complete worked solution
(a)
$x^{1/2+3/2} = x^2$
(b)
$x^{5/3-2/3} = x^1 = x$
(c)
$x^{2}$
(d)
$x^2y^8$
QUESTION 7
7 marks
Criterion D
Hard
a.
Solve $x^{1/2} = 7$ for $x$ (square both sides).
[2] b.
Solve $x^{1/3} = 4$ for $x$.
[2] c.
Solve $x^{2/3} = 9$ for $x$.
[3] Show complete worked solution
(a)
$x = 7^2 = 49$
(b)
$x = 4^3 = 64$
(c)
$\left(x^{1/3}\right)^2 = 9 \Rightarrow x^{1/3} = 3 \Rightarrow x = 3^3 = 27$
QUESTION 8
4 marks
Criterion A
Medium
a.
Evaluate $100^{1/2}$.
[1] b.
Evaluate $1000^{1/3}$.
[1] c.
Evaluate $81^{1/4}$.
[1] d.
Evaluate $1^{99/100}$.
[1] Show complete worked solution
(a)
$10$
(b)
$10$
(c)
$3$
(d)
$1$
QUESTION 9
8 marks
Criterion C
Hard
a.
Evaluate $32^{-1/5}$.
[2] b.
Evaluate $\left(\dfrac{1}{4}\right)^{1/2}$.
[2] c.
Evaluate $\left(\dfrac{8}{27}\right)^{1/3}$.
[2] d.
Evaluate $\left(\dfrac{9}{4}\right)^{-1/2}$.
[2] Show complete worked solution
(a)
$\left(32^{1/5}\right)^{-1} = 2^{-1} = \dfrac{1}{2}$
(b)
$\dfrac{1}{2}$
(c)
$\dfrac{8^{1/3}}{27^{1/3}} = \dfrac{2}{3}$
(d)
$\left(\dfrac{4}{9}\right)^{1/2} = \dfrac{2}{3}$
QUESTION 10
6 marks
Criterion D
Medium
The volume $V$ of a cube with side length $s$ is $V = s^3$, so the side length can be found using $s = V^{1/3}$. The area $A$ of a square is $A = s^2$, so its side length is $s = A^{1/2}$.
a.
A cube has volume $125$ cm³. Determine its side length.
[2] b.
A different cube has volume $512$ cm³. Determine its side length.
[2] c.
A square has area $196$ cm². Determine its side length.
[2] Show complete worked solution
(a)
$125^{1/3} = 5$ cm
(b)
$512^{1/3} = 8$ cm
(c)
$196^{1/2} = 14$ cm
QUESTION 11
6 marks
Criterion B
Hard
Consider the values $9^{1/2}$, $9^0$, $9^{-1/2}$, and $9^1$.
a.
Evaluate all four values.
[3] b.
Arrange the four values in ascending order.
[1] c.
Describe the relationship between the exponent and the size of the value, for a base greater than $1$.
[2] Show complete worked solution
(a)
$9^{1/2}=3$, $9^0=1$, $9^{-1/2}=\dfrac{1}{3}$, $9^1=9$
(b)
$9^{-1/2} < 9^0 < 9^{1/2} < 9^1$, i.e. $\dfrac{1}{3} < 1 < 3 < 9$
(c)
For a base greater than $1$, increasing the exponent increases the value: negative exponents give values less than $1$, an exponent of $0$ gives exactly $1$, and positive exponents give values greater than $1$, growing larger as the exponent increases.
QUESTION 12
5 marks
Criterion C
Hard
Simplify $\dfrac{x^{3/2} \times x^{1/2}}{x^{-1}}$, giving your answer in simplest index form.
a.
Simplify the numerator, $x^{3/2} \times x^{1/2}$.
[1] b.
Hence simplify $\dfrac{x^{3/2} \times x^{1/2}}{x^{-1}}$ fully.
[2] c.
Hence evaluate the original expression when $x = 4$.
[2] Show complete worked solution
(a)
$x^{3/2+1/2} = x^2$
(b)
$\dfrac{x^2}{x^{-1}} = x^3$
(c)
$4^3 = 64$