OCR MEI C1 (Core Mathematics 1)

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]

1. Simplify \((5a^2b)^2 \times 2b^4\). [2]
2. Evaluate \(\left(\frac{4}{16}\right)^{-1}\). [1]
3. Evaluate \((16)^{\frac{3}{2}}\). [2]

1. Find the value of \(144^{-\frac{1}{2}}\). [2]
2. Simplify \(\frac{1}{5 + \sqrt{7}} + \frac{4}{5 - \sqrt{7}}\). Give your answer in the form \(\frac{a + b\sqrt{7}}{c}\). [3]

Find the value of each of the following.

1. \(5^2 \times 5^{-2}\) [2]
2. \(100^{\frac{1}{2}}\) [1]

State the value of each of the following.

1. \(2^{-3}\) [1]
2. \(9^0\) [1]

1. Express \(125\sqrt{5}\) in the form \(5^k\). [2]
2. Simplify \((4a^3b^5)^2\). [2]

1. Find the value of \(\left(\frac{1}{25}\right)^{-\frac{1}{2}}\). [2]
2. Simplify \(\frac{(2x^2y^3z)^5}{4y^5z}\). [3]

1. Write down the value of \(\left(\frac{1}{4}\right)^0\). [1]
2. Find the value of \(16^{-\frac{3}{2}}\). [3]

1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
2. Find the value of \(\left(\frac{1}{2}\right)^{-5}\). [2]

Find the value of each of the following, giving each answer as an integer or fraction as appropriate.

1. \(8^{\frac{1}{3}}\) [2]
2. \(\left(\frac{7}{3}\right)^{-2}\) [2]

1. Simplify \(6\sqrt{2} \times 5\sqrt{3} \times \sqrt{24}\). [2]
2. Express \((2 - 3\sqrt{5})^2\) in the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are integers. [3]

Simplify the following.

1. \(\frac{16^{\frac{1}{3}}}{81^{\frac{1}{4}}}\) [2]
2. \(\frac{12(a^3b^2c)^4}{4a^2c^6}\) [3]