1.02a Indices: laws of indices for rational exponents

230 questions

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OCR C1 Q4
6 marks Easy -1.2
4. (i) Evaluate $$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (ii) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
OCR C1 Q6
7 marks Moderate -0.5
  1. (i) Given that \(y = x ^ { \frac { 1 } { 3 } }\), show that the equation
$$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$ can be rewritten as $$2 y ^ { 2 } - 7 y + 3 = 0 .$$ (ii) Hence, solve the equation $$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$
OCR C1 Q6
7 marks Moderate -0.3
6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$
  1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
  2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).
OCR C1 Q6
7 marks Moderate -0.8
  1. (i) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
    (ii) Find the value of \(x\) such that
$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
OCR MEI C1 Q3
3 marks Moderate -0.8
3 Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac { 1 } { 3 } \pi r ^ { 2 } ( a + b )$$
OCR MEI C1 Q4
5 marks Moderate -0.8
4
  1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
  2. Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).
OCR MEI C1 Q6
3 marks Easy -1.8
6 Make \(b\) the subject of the following formula. $$a = \frac { 2 } { 3 } b ^ { 2 } c$$
OCR MEI C1 Q8
4 marks Moderate -0.5
8 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.
OCR MEI C1 Q13
3 marks Easy -1.8
13 Make \(v\) the subject of the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\).
OCR MEI C1 Q14
3 marks Easy -1.8
14 Make \(t\) the subject of the formula \(s = \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q16
3 marks Easy -1.8
16 The volume of a cone is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). Make \(r\) the subject of this formula.
OCR MEI C2 2008 January Q7
5 marks Easy -1.3
7
  1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
  2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
  3. Sketch the curve with equation \(y = 2 ^ { x }\).
OCR MEI C2 Q6
5 marks Standard +0.3
6 Find the solution to this equation, correct to 3 significant figures. $$\left( 2 ^ { x } \right) \left( 2 ^ { x + 1 } \right) = 10 .$$
OCR C2 Q2
6 marks Standard +0.3
2. Given that $$y = 2 x ^ { \frac { 3 } { 2 } } - 1 ,$$ find $$\int y ^ { 2 } \mathrm {~d} x .$$
Edexcel C1 2014 June Q2
4 marks Easy -1.3
2.(a)Write down the value of \(32 ^ { \frac { 1 } { 5 } }\) (b)Simplify fully \(\left( 32 x ^ { 5 } \right) ^ { - \frac { 2 } { 5 } }\) I敖
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OCR C1 2009 January Q2
4 marks Easy -1.3
2 Simplify
  1. \(( \sqrt [ 3 ] { x } ) ^ { 6 }\),
  2. \(\frac { 3 y ^ { 4 } \times ( 10 y ) ^ { 3 } } { 2 y ^ { 5 } }\).
OCR C1 2009 January Q3
5 marks Standard +0.3
3 Solve the equation \(3 x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } - 2 = 0\).
OCR C1 2010 January Q4
7 marks Easy -1.3
4 Solve the equations
  1. \(3 ^ { m } = 81\),
  2. \(\left( 36 p ^ { 4 } \right) ^ { \frac { 1 } { 2 } } = 24\),
  3. \(5 ^ { n } \times 5 ^ { n + 4 } = 25\).
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 2012 January Q4
5 marks Easy -1.8
4 Evaluate
  1. \(3 ^ { - 2 }\),
  2. \(16 ^ { \frac { 3 } { 4 } }\),
  3. \(\frac { \sqrt { 200 } } { \sqrt { 8 } }\).
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 2011 June Q3
5 marks Easy -1.3
3 Simplify
  1. \(\frac { ( 4 x ) ^ { 2 } \times 2 x ^ { 3 } } { x }\),
  2. \(\left( 36 x ^ { - 2 } \right) ^ { - \frac { 1 } { 2 } }\).
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 2015 June Q4
5 marks Moderate -0.3
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } - 6 = 0\).