1.02a Indices: laws of indices for rational exponents

230 questions

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OCR MEI AS Paper 1 Specimen Q1
2 marks Easy -1.2
1 Simplify \(\frac { \left( 2 x ^ { 2 } y \right) ^ { 3 } \times 4 x ^ { 3 } y ^ { 5 } } { 2 x y ^ { 10 } }\).
OCR MEI AS Paper 2 2019 June Q1
3 marks Easy -1.2
1 Solve the equation \(4 x ^ { - \frac { 1 } { 2 } } = 7\), giving your answer as a fraction in its lowest terms.
OCR MEI Paper 1 2020 November Q1
2 marks Easy -1.3
1 Simplify \(\left( \frac { 27 } { x ^ { 9 } } \right) ^ { \frac { 2 } { 3 } } \times \left( \frac { x ^ { 4 } } { 9 } \right)\).
[0pt] [2]
OCR MEI Paper 2 2023 June Q3
3 marks Moderate -0.8
3 In this question you must show detailed reasoning.
Find the smallest possible positive integers \(m\) and \(n\) such that \(\left( \frac { 64 } { 49 } \right) ^ { - \frac { 3 } { 2 } } = \frac { m } { n }\).
OCR MEI Paper 3 2020 November Q9
3 marks Standard +0.3
9
  1. Show that if \(a = 1\) and \(b > 1\) then \(\mathrm { a } ^ { \mathrm { b } } < \mathrm { b } ^ { \mathrm { a } }\).
  2. Find integer values of \(a\) and \(b\) with \(b > a > 1\) and \(\mathrm { a } ^ { \mathrm { b } }\) not greater than \(\mathrm { b } ^ { \mathrm { a } }\) (a counter example to the conjecture given in lines 7-8).
AQA C1 2016 June Q2
5 marks Easy -1.3
2
  1. Simplify \(( 3 \sqrt { 5 } ) ^ { 2 }\).
  2. Express \(\frac { ( 3 \sqrt { 5 } ) ^ { 2 } + \sqrt { 5 } } { 7 + 3 \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
    [0pt] [4 marks]
Edexcel C1 Q1
5 marks Easy -1.2
  1. 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 \(y\),
    2. calculate the exact value of \(2 ^ { y - x }\).
    3. \(f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0\).
    4. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
    5. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
    6. The sum of an arithmetic series is \(\sum _ { r = 1 } ^ { n } ( 80 - 3 r )\).
    7. Write down the first two terms of the series.
    8. Find the common difference of the series.
    Given that \(n = 50\),
  2. find the sum of the series.
Edexcel C1 Q4
6 marks Moderate -0.8
4. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
(b) 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.
Edexcel C1 Q6
7 marks Moderate -0.3
6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } } .$$
  1. Evaluate 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 \(k \sqrt { 2 }\).
Edexcel C1 Q4
6 marks Easy -1.3
  1. (a) Evaluate
$$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (b) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
Edexcel C1 Q4
6 marks Easy -1.8
4. (a) Solve the inequality $$4 ( x - 2 ) < 2 x + 5$$ (b) Find the value of \(y\) such that $$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$
AQA C2 2005 January Q4
9 marks Moderate -0.8
4
  1. Write \(\sqrt { x }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
  2. Hence express \(\sqrt { x } ( x - 1 )\) in the form \(x ^ { p } - x ^ { q }\).
  3. Find \(\int \sqrt { x } ( x - 1 ) \mathrm { d } x\).
  4. Hence show that \(\int _ { 1 } ^ { 2 } \sqrt { x } ( x - 1 ) \mathrm { d } x = \frac { 4 } { 15 } ( \sqrt { 2 } + 1 )\).
AQA C2 2010 January Q2
7 marks Moderate -0.8
2 At the point \(( x , y )\) on a curve, where \(x > 0\), the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$
  1. Write \(\sqrt { x ^ { 5 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
  2. Find \(\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x\).
  3. Hence find the equation of the curve, given that the curve passes through the point \(( 1,3 )\).
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 \(\sqrt { 2 } \times 2 ^ { x } = \frac { 1 } { 8 }\).
AQA C2 2012 January Q3
3 marks Easy -1.2
3
  1. Write \(\sqrt [ 4 ] { x ^ { 3 } }\) in the form \(x ^ { k }\).
  2. Write \(\frac { 1 - x ^ { 2 } } { \sqrt [ 4 ] { x ^ { 3 } } }\) in the form \(x ^ { p } - x ^ { q }\).
AQA C2 2008 June Q1
9 marks Moderate -0.8
1
  1. Write \(\sqrt { x ^ { 3 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
    (1 mark)
  2. A curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find the equation of the tangent to the curve at the point where \(x = 4\), giving your answer in the form \(y = m x + c\).
AQA C2 2010 June Q6
13 marks Moderate -0.8
6 A curve \(C\) has the equation $$y = \frac { x ^ { 3 } + \sqrt { x } } { x } , \quad x > 0$$
  1. Express \(\frac { x ^ { 3 } + \sqrt { x } } { x }\) in the form \(x ^ { p } + x ^ { q }\).
    1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the normal to the curve \(C\) at the point on the curve where \(x = 1\).
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence deduce that the curve \(C\) has no maximum points. \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-7_1463_1707_1244_153}
Edexcel C2 Q6
11 marks Standard +0.3
6. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0\),
  1. find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
  2. Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
  3. Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
Edexcel C1 Q1
Easy -1.3
  1. (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
    (b) Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
  2. Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
    (a) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
    (b) find \(\int y \mathrm {~d} x\).
AQA C2 2007 January Q3
5 marks Easy -1.3
3
  1. Write down the values of \(p , q\) and \(r\) given that:
    1. \(64 = 8 ^ { p }\);
    2. \(\frac { 1 } { 64 } = 8 ^ { q }\);
    3. \(\sqrt { 8 } = 8 ^ { r }\).
  2. Find the value of \(x\) for which $$\frac { 8 ^ { x } } { \sqrt { 8 } } = \frac { 1 } { 64 }$$
AQA C2 2007 June Q1
8 marks Easy -1.3
1
  1. Simplify:
    1. \(x ^ { \frac { 3 } { 2 } } \times x ^ { \frac { 1 } { 2 } }\);
    2. \(x ^ { \frac { 3 } { 2 } } \div x\);
    3. \(\left( x ^ { \frac { 3 } { 2 } } \right) ^ { 2 }\).
    1. Find \(\int 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { 1 } ^ { 9 } 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
AQA AS Paper 2 2021 June Q1
1 marks Easy -1.8
1 Express as a single power of \(a\) $$\frac { a ^ { 2 } } { \sqrt { a } }$$ where \(a \neq 0\) Circle your answer. \(a ^ { 1 }\) \(a ^ { \frac { 3 } { 2 } }\) \(a ^ { \frac { 5 } { 2 } }\) \(a ^ { 4 }\)
Edexcel PURE 2024 October Q2
Easy -1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
  2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.
Pre-U Pre-U 9794/2 2012 Specimen Q1
7 marks Easy -1.8
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
Pre-U Pre-U 9794/2 2016 Specimen Q1
9 marks Easy -1.3
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).