## 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 |

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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