1.02a Indices: laws of indices for rational exponents

230 questions

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Edexcel P1 2018 Specimen Q2
5 marks Easy -1.2
  1. Given that \(3^{-1.5} = a\sqrt{3}\) find the exact value of \(a\) [2]
  2. Simplify fully \(\frac{(2x^{\frac{1}{2}})^3}{4x^2}\) [3]
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(16^{-1}\). [1]
  2. Find the value of \(16^{-\frac{1}{2}}\). [2]
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(8^{-1}\). [1]
  2. Find the value of \(8^{-\frac{2}{3}}\). [2]
Edexcel C1 Q7
8 marks Moderate -0.8
  1. Show that \(\frac{(3 - \sqrt{x})^2}{\sqrt{x}}\) can be written as \(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\). [2]
Given that \(\frac{dy}{dx} = \frac{(3 - \sqrt{x})^2}{\sqrt{x}}\), \(x > 0\), and that \(y = \frac{2}{3}\) at \(x = 1\),
  1. find \(y\) in terms of \(x\). [6]
Edexcel C1 Q2
4 marks Easy -1.8
  1. Find the value of \(8^{-1}\). [2]
  2. Simplify \(\frac{15x^4}{3x}\). [2]
Edexcel C1 Q5
5 marks Moderate -0.8
  1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
  2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]
Edexcel C1 Q9
5 marks Moderate -0.8
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]
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]
Edexcel C1 Q1
5 marks Easy -1.2
  1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
  2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]
OCR C1 2013 January Q2
6 marks Easy -1.3
Solve the equations
  1. \(3^n = 1\), [1]
  2. \(t^{-3} = 64\), [2]
  3. \((8p^6)^{\frac{1}{3}} = 8\). [3]
OCR C1 2006 June Q2
6 marks Easy -1.2
  1. Evaluate \(27^{\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1 - \sqrt{5}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]
OCR C1 2013 June Q1
4 marks Easy -1.3
Express each of the following in the form \(a\sqrt{5}\), where \(a\) is an integer.
  1. \(4\sqrt{15} \times \sqrt{3}\) [2]
  2. \(\frac{20}{\sqrt{5}}\) [1]
  3. \(5^{\frac{3}{2}}\) [1]
OCR C1 2014 June Q2
5 marks Easy -1.3
Express each of the following in the form \(k\sqrt{3}\), where \(k\) is an integer.
  1. \(\frac{6}{\sqrt{3}}\) [1]
  2. \(10\sqrt{3} - 6\sqrt{27}\) [2]
  3. \(3^{\frac{3}{2}}\) [2]
OCR MEI C1 Q2
3 marks Easy -1.2
Make \(r\) the subject of \(V = \frac{4}{3}\pi r^3\). [3]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of the following.
  1. \(\left(\frac{1}{3}\right)^{-2}\) [2]
  2. \(16^{\frac{1}{4}}\) [2]
OCR MEI C1 2006 June Q1
3 marks Easy -1.2
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. [3]
OCR MEI C1 2006 June Q9
5 marks Easy -1.3
Simplify the following.
  1. \(\frac{16^{\frac{1}{4}}}{81^{\frac{1}{4}}}\) [2]
  2. \(\frac{12(a^3b^2c)^4}{4a^2c^6}\) [3]
OCR MEI C1 2009 June Q2
3 marks Easy -1.8
Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]
OCR MEI C1 2009 June Q7
3 marks Easy -1.8
Find the value of each of the following.
  1. \(5^2 \times 5^{-2}\) [2]
  2. \(100^{\frac{1}{2}}\) [1]
OCR MEI C1 2010 June Q2
5 marks Easy -1.8
  1. Simplify \((5a^2b)^3 \times 2b^4\). [2]
  2. Evaluate \(\left(\frac{1}{16}\right)^{-1}\). [1]
  3. Evaluate \((16)^{\frac{1}{2}}\). [2]
OCR MEI C1 2011 June Q3
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{9}{16}\right)^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(2ac^2)^3 \times 9a^2c}{36a^4c^{12}}\). [3]
OCR MEI C1 2012 June Q2
3 marks Easy -1.8
Make \(b\) the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]
OCR MEI C1 2012 June Q3
4 marks Easy -1.8
  1. Evaluate \(\left(\frac{1}{5}\right)^{-2}\). [2]
  2. Evaluate \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\). [2]
OCR MEI C1 2013 June Q3
5 marks Easy -1.8
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{1}{4}}\). [3]
OCR MEI C1 2013 June Q4
3 marks Easy -1.2
Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac{1}{3}\pi r^2(a + b)$$ [3]