 # Exam-Style Questions on Indices

## Problems on Indices adapted from questions set in previous Mathematics exams.

### 1.

GCSE Higher

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

### 2.

GCSE Higher

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

### 3.

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.

### 4.

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

Work out the exact value of $$n$$.

### 5.

GCSE Higher

Without using a calculator find the values of the following:

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

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

### 6.

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.

### 7.

GCSE Higher

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

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

### 8.

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 $$x = 4$$

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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