Express in form with given base

A question is this type if and only if it asks to rewrite an expression as a power of a specified base, such as expressing 125√5 in the form 5^k or 8^(2x+3) in the form 2^y.

19 questions · Easy -1.4

1.02a Indices: laws of indices for rational exponents
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Edexcel C12 2019 June Q2
6 marks Moderate -0.8
  1. Find the value of \(a\) and the value of \(b\) for which \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } \equiv 2 ^ { a x + b }\)
  2. Hence solve the equation \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } = 2 \sqrt { 2 }\)
Edexcel C1 2009 June Q2
3 marks Easy -1.8
2. Given that \(32 \sqrt { } 2 = 2 ^ { a }\), find the value of \(a\).
Edexcel C1 2016 June Q2
2 marks Easy -1.3
Express \(9 ^ { 3 x + 1 }\) in the form \(3 ^ { y }\), giving \(y\) in the form \(a x + b\), where \(a\) and \(b\) are constants.
(2)
Edexcel C1 2013 January Q2
2 marks Easy -1.3
Express \(8 ^ { 2 x + 3 }\) in the form \(2 ^ { y }\), stating \(y\) in terms of \(x\).
OCR C1 2008 June Q1
4 marks Easy -1.8
1 Express each of the following in the form \(4 ^ { n }\) :
  1. \(\frac { 1 } { 16 }\),
  2. 64 ,
  3. 8 .
OCR MEI C1 2009 January Q7
4 marks Easy -1.3
7
  1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
  2. Simplify \(\left( 4 a ^ { 3 } b ^ { 5 } \right) ^ { 2 }\).
OCR C1 2011 January Q3
5 marks Easy -1.8
3 Express each of the following in the form \(8 ^ { p }\) :
  1. \(\sqrt { 8 }\),
  2. \(\frac { 1 } { 64 }\),
  3. \(2 ^ { 6 } \times 2 ^ { 2 }\).
OCR C1 2009 June Q3
4 marks Easy -1.8
3 Express each of the following in the form \(3 ^ { n }\) :
  1. \(\frac { 1 } { 9 }\),
  2. \(\sqrt [ 3 ] { 3 }\),
  3. \(3 ^ { 10 } \times 9 ^ { 15 }\).
OCR C1 2012 June Q2
5 marks Easy -1.3
2 Express each of the following in the form \(7 ^ { k }\) :
  1. \(\sqrt [ 4 ] { 7 }\),
  2. \(\frac { 1 } { 7 \sqrt { 7 } }\),
  3. \(7 ^ { 4 } \times 49 ^ { 10 }\).
OCR C1 2015 June Q3
5 marks Easy -1.3
3 Express each of the following in the form \(5 ^ { k }\).
  1. \(25 ^ { 4 }\)
  2. \(\frac { 1 } { \sqrt [ 4 ] { 5 } }\)
  3. \(( 5 \sqrt { 5 } ) ^ { 3 }\)
OCR C1 2016 June Q5
5 marks Easy -1.3
5 Express the following in the form \(2 ^ { p }\).
  1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
  2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)
OCR MEI C1 2011 January Q7
5 marks Easy -1.2
7
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
AQA C2 2011 January Q2
5 marks Easy -1.3
2
  1. Write down the values of \(p , q\) and \(r\) given that:
    1. \(8 = 2 ^ { p }\);
    2. \(\frac { 1 } { 8 } = 2 ^ { q }\);
    3. \(\sqrt { 2 } = 2 ^ { r }\).
  2. Find the value of \(x\) for which \(\sqrt { 2 } \times 2 ^ { x } = \frac { 1 } { 8 }\).
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),
  1. find the exact value of \(x\) and the exact value of \(y\), [3]
  2. calculate the exact value of \(2^{y-x}\). [2]
OCR MEI C1 2013 June Q7
5 marks Moderate -0.8
  1. Express \(125\sqrt{5}\) in the form \(5^t\). [2]
  2. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
OCR MEI C1 Q1
5 marks Easy -1.2
  1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
  2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]
OCR MEI C1 Q7
4 marks Easy -1.8
  1. Express \(125\sqrt{5}\) in the form \(5^k\). [2]
  2. Simplify \((4a^3b^5)^2\). [2]
AQA AS Paper 2 2020 June Q1
11 marks
Identify the expression below that is equivalent to \(e^{-\frac{2}{5}}\) Circle your answer. [1 mark] \(\frac{1}{\sqrt[5]{e^2}}\) \quad \(-\sqrt{e^5}\) \quad \(-\sqrt[5]{e^2}\) \quad \(\frac{1}{\sqrt{e^5}}\)
SPS SPS SM 2025 November Q1
7 marks Easy -1.3
Do not use a calculator for this question
  1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
    1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
    2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]