Edexcel IGCSE Math A Higher 1.4 Powers and Roots Questions

Practise powers, roots, index laws, prime factor form, HCF, LCM and surds through calculation and show-that questions.

Exam points

apply index laws to products and quotients, including negative powers in a single base
use prime factors to find products of powers, HCF or LCM for integers
manipulate surds by simplifying roots or rationalising denominators into a required form

Covered syllabus points

1.4.A Square numbers and cube numbers 1.4.B Squares, square roots, cubes and cube roots 1.4.C Index notation and index laws 1.4.D Prime factor decomposition 1.4.E Highest common factors (HCF) and lowest common 1.4.HA Meaning of surds 1.4.HB Manipulating and rationalising surds 1.4.HC Fractional and negative indices

[Maximum number: 3]

$12 P=3^{3} \times 5^{2} \times 7$

Q=3^{2} \times 5 \times 7^{2}

(a)

Write down the highest common factor (HCF) of P and Q

\begin{aligned} & P=3^{3} \times 5^{2} \times 7 \\ & Q=3^{2} \times 5 \times 7^{2} \end{aligned}

[ 1 ]

Final answer:
$3^{2}\times5\times7$

Accept $3\times3\times5\times7$ oe or 315.

(b)

Work out the value of $P^{3} \times Q$

Give your answer in the form $3^{x} \times 5^{y} \times 7^{z}$ where x, y and z are positive integers.

[ 2 ]

$P^{3}Q=3^{11}\times5^{7}\times7^{5}$

Final answer:
$3^{11}\times5^{7}\times7^{5}$

B1 for an answer in the form $3^{p}\times5^{q}\times7^{r}$ where one or two exponents are correct.

[Maximum number: 3]

$9 \quad N=480 \times 10^{9}$

Write N as a product of powers of its prime factors. Show your working clearly.

$N=2^{14}\times3\times5^{10}$

Final answer:
$2^{14}\times3\times5^{10}$
B2 for an answer in the form $2^m\times3\times5^n$ , where m and n are positive integers.

[Maximum number: 3]

Show that $\frac{2}{6-3 \sqrt{2}}$ can be written in the form $\frac{a+\sqrt{a}}{b}$ where a and b are integers.
Show your working clearly.

E.g.
$\begin{aligned} \frac{2}{6-3 \sqrt{2}} \times \frac{6+3 \sqrt{2}}{6+3 \sqrt{2}} \text { or } \frac{2}{6-3 \sqrt{2}} \times \frac{-6-3 \sqrt{2}}{-6-3 \sqrt{2}} \end{aligned}$

M1 for rationalising the denominator by
multiplying numerator and denominator
by $6+3 \sqrt{2}$ (or $-6-3 \sqrt{2}$ )

$\begin{aligned} \frac{12+6 \sqrt{2}}{36-18 \sqrt{2}+18 \sqrt{2}-18} \text { or } \frac{12+6 \sqrt{2}}{18} \text { or } \frac{12+6 \sqrt{2}}{6^{2}-(3 \sqrt{2})^{2}} \text { or } \frac{12+6 \sqrt{2}}{6^{2}-9 \times 2} \end{aligned}$

M1 (numerator may be expanded or
denominator may be 4 terms which need
to be all correct)

$\frac{2+\sqrt{2}}{3}$

A1 or for stating a=2 and b=3
for $\frac{2+\sqrt{2}}{3}$ from correct working
dep on M2

[Maximum number: 2]

$12 P=3^{3} \times 5^{2} \times 7$

Q=3^{2} \times 5 \times 7^{2}

Work out the value of $P^{3} \times Q$

Give your answer in the form $3^{x} \times 5^{y} \times 7^{z}$ where x, y and z are positive integers.

$P^{3}Q=3^{11}\times5^{7}\times7^{5}$

Final answer:
$3^{11}\times5^{7}\times7^{5}$

B1 for an answer in the form $3^{p}\times5^{q}\times7^{r}$ where one or two exponents are correct.

[Maximum number: 1]

$12 P=3^{3} \times 5^{2} \times 7$

Q=3^{2} \times 5 \times 7^{2}

Write down the highest common factor (HCF) of P and Q

\begin{aligned} & P=3^{3} \times 5^{2} \times 7 \\ & Q=3^{2} \times 5 \times 7^{2} \end{aligned}

Final answer:
$3^{2}\times5\times7$

Accept $3\times3\times5\times7$ oe or 315.

[Maximum number: 2]

$7 \quad \frac{3^{-2} \times 3^{5}}{3^{10}}=3^{n}$
Find the value of n

Index calculation:
-2+5-10=-7

Final answer:
n=-7

Allow $3^{-7}$ .

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question 4

Question 5

Question 6