1.02a Indices: laws of indices for rational exponents

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]

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]

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]

Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),

1. find, to 3 significant figures, the value of x for which f(x) = 5. [3]
2. Show that f(x) may be written in the form \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]

Identify the expression below which is equivalent to \(\left(\frac{2x}{5}\right)^{-3}\) Circle your answer. [1 mark] \(\frac{8x^3}{125}\) \quad \(\frac{125x^3}{8}\) \quad \(\frac{125}{8x^3}\) \quad \(\frac{8}{125x^3}\)

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 \(\frac{1}{\sqrt{e^5}}\)

Simplify \(\sqrt{a^{\frac{2}{3}} \times a^{\frac{2}{5}}}\) Circle your answer. [1 mark] \(a^{\frac{2}{15}}\) \quad \(a^{\frac{4}{15}}\) \quad \(a^{\frac{8}{15}}\) \quad \(a^{\frac{16}{15}}\)

Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]

Express $$5 - \frac{\sqrt[3]{x}}{x^2}$$ in the form $$5x^p - x^q$$ where \(p\) and \(q\) are constants. [2 marks]

A student was asked to give the exact solution to the equation $$2^{2x+4} - 9(2^x) = 0$$ The student's attempt is shown below: $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ Let \(2^x = y\) $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$

1. Identify the two errors made by the student. [2]
2. Find the exact solution to the equation. [2]

In this question you must show detailed reasoning.

1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]

Simplify the expression \(\sqrt[3]{512a^7} - \frac{a^{\frac{7}{2}} \times a^{-\frac{1}{3}}}{a^6}\). [4]

Simplify fully the following expressions:

1. \(\frac{7y^{13}}{35y^7}\) [1]
2. \(6x^{-2} \div x^{-5}\) [1]

In this question you must show detailed reasoning.

1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]

Simplify \(\left(\frac{x^{12}}{16}\right)^{-\frac{3}{4}}\) [2]