Numbers - Indices, Powers & Roots
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calculator Simple Total: 4 marks

\[ p = 2 \times 10^3 \]

\[ q = 8 \times 10^{-5} \]

Work out the following, giving each answer in standard form.

(a)
\(pq\)
(b)
\(\frac{p}{q}\)
Answer
(a)
\(1.6 \times 10^{-1}\)
(b)
\(2.5 \times 10^{7}\)
calculator Simple Total: 1 mark

Calculate

\[ \frac{\sqrt[3]{16}}{1.3^2} \]

Answer
\( \approx 1.49 \)
non-calculator Simple Total: 1 mark
\[\text{Find the value of } 12^0\]
Answer
\(12^0 = 1\)
non-calculator Simple Total: 1 mark
Write the following as a single power.
(a)
\(2^3 \times 2^6\)
(b)
$$ \frac{3^9}{3^4} $$
Answer
(a)
\(2^9\)
(b)
\(3^5\)
non-calculator Simple Total: 1 mark
\[ \text{Write } 5^{17} \times 5^2 \text{ as a single power of 5.} \]
Answer
\[ 5^{19} \]
non-calculator Simple Total: 1 mark
Simplify leaving your answer in index form
(a)
\(3^5 \times 3^3 \times 3\)
(b)
\((5^7)^2\)
(c)
\(5^9 \div 5^3\)
Answer
(a)
\[ \text{(a) } 3^9 \]
(b)
\[ \text{(b) } 5^{14} \]
(c)
\[ \text{(c) } 5^6 \]
non-calculator Simple Total: 1 mark
Work out the value of

\[ \frac{2^9 \times 2^{-2}}{2^4} \]

Answer
\(2^3 = 8\)
non-calculator Simple Total: 3 marks
Write as a single power of 7
(a)
\(7^8 \div 7^3\)
[1]
(b)
\(\frac{7^4 \times 7^5}{7}\)
[1]
(c)
\(49 \times 7\)
[1]
Answer
(a)
\(7^5\) [1]
(b)
\(7^8\)
(c)
\(7^3\)
non-calculator Medium Total: 2 marks

Write

\(64 \times 4^5\) as a single power of 4

[2]
Answer
\(4^8\)
non-calculator Medium Total: 2 marks

Work out the value of

\[ \left(2^{\frac{7}{2}} - 2^{\frac{1}{2}}\right)^2 \]

Answer
\(32\)
non-calculator Medium Total: 2 marks

\[ \left(\frac{3}{8}\right)^{\frac{3}{8}} \times \left(\frac{3}{8}\right)^{\frac{1}{8}} = p^q \]

Find the value of \(p\) and the value of \(q\).

Answer:

\(p = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)

\(q = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)

Answer
\(p = \frac{3}{8}\) \(q = \frac{1}{2}\)
non-calculator Medium Total: 2 marks

Find the value of

\[ \left(\frac{1}{4}\right)^{0.5} \]

Answer
\(\frac{1}{2}\)
non-calculator Hard Total: 2 marks

Work out

\[ 16^{\frac{1}{4}} \]

Answer
\(2\)
non-calculator Hard Total: 1 mark

Find the value of

\[ 81^{\frac{3}{4}} \]

[2]
Answer
\(27\)
non-calculator Hard Total: 5 marks
Simplify leaving your answers as fractions.
(a)
\(2^{-3}\)
[1]
(b)
\(\left(\frac{27}{343}\right)^{\frac{1}{3}}\)
[2]
(c)
\(\left(\sqrt{\frac{3}{8}}\right)^4\)
[2]
Answer
(a)
\(\frac{1}{8}\)
(b)
\(\frac{3}{7}\)
(c)
\(\frac{9}{64}\)
non-calculator Hard Total: 1 mark

\[ 5^{-3} + 5^{-4} = k \times 5^{-4} \]

Find the value of \(k\).

[2]
Answer
\(k = 6\)
non-calculator Hard Total: 2 marks

Find the exact value of

\[ 8^{\frac{2}{3}} \times 49^{-\frac{1}{2}} \]

Answer
\(\frac{4}{7}\)
non-calculator Hard Total: 2 marks

Find the exact value of

\[ 8^{\frac{2}{3}} \times 49^{-\frac{1}{2}} \]

Answer
\(\frac{4}{7}\)