Exam-Style Questions on Indices

1.	GCSE Higher

Show that 206 can be written as the sum of a power of five and a square number.

Worked Solution

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2.	GCSE Higher

Simplify the following expressions:

(a)

$$ a^7 \times a^8 $$

(b)

$$ 7b^9 + 8b^9 $$

(c)

$$ (2c)^5 \times c^{-3} $$

(d)

$$ (d^4 e^3) \div (d^{-1} e^3) $$

Worked Solution

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3.	GCSE Higher

Without using a calculator, show clearly that \(27^{\frac23}\) is equal to \(9\).

Worked Solution

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4.	GCSE Higher

There's an old Elvish wives tale that roughly translated reads:

Four To The Half Power Is Simple To Do

Just Halve The Four And The Answer Is Two.

Modern-day Elves extend this method to calculate other powers such as

$$ 16^{ \frac{1}{4}} = \frac{1}{4} \text{ of } 16 = 4 $$

Is this calculation correct? If it is not explain what is wrong.

Worked Solution

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5.	GCSE Higher

$$ 9^{\frac{1}{3}} \times 3^n = 27^{\frac{4}{5}} $$

Work out the exact value of \(n\).

Worked Solution

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6.	GCSE Higher

Without using a calculator find the values of the following:

(a) \(25^{-\frac12} \)

(b) \( \left( \frac{27}{64} \right)^{ \frac23} \)

Worked Solution

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7.	GCSE Higher

(a) The \(n\)th term of a sequence is \(2^n+2^{n+1}\)

Work out the 8th term of the sequence.

(b) The \(n\)th term of a different sequence is \(9(3^n + 3^{n+1})\)

Expand and express this expression as the sum of two powers of three.

Worked Solution

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8.	GCSE Higher

If a, b and c are positive integers use the following statements to find the values of a, b and c.

$$ (ab^c)^3 = 27b^{21} $$ $$ b= 9a $$

Worked Solution

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9.	GCSE Higher

\(y = a \times b^{x – 2}\) where \(a\) and \(b\) are numbers.

\(y = 5\) when \(x = 2\)

\(y = 0.005\) when \(x = 5\)

Work out the value of \(y\) when \(x = 4\)

Worked Solution

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