A-Level Maths | Indices
Rules of Indices
$x^{\frac{1}{n}} = \sqrt[n]{x}$
$x^{\frac{m}{n}} = \left(\sqrt[n]{x}\right)^m$
$x^{-n} = \dfrac{1}{x^n}$
Question 1
Solve each equation.
a) $x^{\frac{1}{2}} = 6$
b) $x^{\frac{1}{3}} = 5$
c) $x^{-\frac{1}{2}} = 2$
d) $x^{-\frac{1}{4}} = \tfrac{1}{3}$
e) $x^{\frac{3}{2}} = 8$
f) $x^{\frac{2}{3}} = 16$
g) $x^{\frac{4}{3}} = 81$
h) $x^{-\frac{3}{2}} = 27$
Show Answers
a) $x = 36$ b) $x = 125$ c) $x = \tfrac{1}{4}$ d) $x = 81$
e) $x = 4$ f) $x = 64$ g) $x = 27$ h) $x = \tfrac{1}{9}$
Question 2
Express in the form $x^k$.
a) $\sqrt{x}$
b) $\dfrac{1}{\sqrt[3]{x}}$
c) $x^2 \times \sqrt{x}$
d) $\dfrac{\sqrt[4]{x}}{x}$
e) $\sqrt{x^3}$
f) $\sqrt{x} \times \sqrt[3]{x}$
g) $(\sqrt{x})^5$
h) $\sqrt[3]{x^2} \times (\sqrt{x})^3$
Show Answers
a) $x^{\frac{1}{2}}$ b) $x^{-\frac{1}{3}}$ c) $x^{\frac{5}{2}}$ d) $x^{-\frac{3}{4}}$
e) $x^{\frac{3}{2}}$ f) $x^{\frac{5}{6}}$ g) $x^{\frac{5}{2}}$ h) $x^{\frac{13}{6}}$
Question 3
Express each of the following in the form $ax^b$, where $a$ and $b$ are rational constants.
a) $\dfrac{4}{\sqrt{x}}$
b) $\dfrac{1}{2x}$
c) $\dfrac{3}{4x^3}$
d) $\dfrac{1}{(3x)^2}$
e) $\dfrac{2}{5\sqrt[3]{x}}$
f) $\dfrac{1}{\sqrt{9x^3}}$
show Answers
a) $4x^{-\frac{1}{2}}$ b) $\tfrac{1}{2}x^{-1}$ c) $\tfrac{3}{4}x^{-3}$
d) $\tfrac{1}{9}x^{-2}$ e) $\tfrac{2}{5}x^{-\frac{1}{3}}$ f) $\tfrac{1}{3}x^{-\frac{3}{2}}$
Question 4
Simplify each of the following.
a) $\dfrac{x^3 + 2x}{x}$
b) $\dfrac{4t^5 - 6t^3}{2t^2}$
c) $\dfrac{x^{\frac{3}{2}} - 3x}{x^{\frac{1}{2}}}$
d) $\dfrac{y^2(y^3 - 6)}{3y}$
e) $\dfrac{p + p^{\frac{3}{2}}}{p^{\frac{3}{4}}}$
f) $\dfrac{8w - 2w^{\frac{1}{2}}}{4w^{-\frac{1}{2}}}$
g) $\dfrac{x + 1}{x^{\frac{1}{2}} + x^{-\frac{1}{2}}}$
h) $\dfrac{2t^3 - 4t}{t^{\frac{3}{2}} - 2t^{-\frac{1}{2}}}$
Show Answers
a) $x^2 + 2$ b) $2t^3 - 3t$ c) $x - 3x^{\frac{1}{2}}$ d) $\tfrac{1}{3}y^4 - 2y$
e) $p^{\frac{1}{4}} + p^{\frac{3}{4}}$ f) $2w^{\frac{3}{2}} - \tfrac{1}{2}w$ g) $x^{\frac{1}{2}}$ h) $2t^{\frac{3}{2}}$
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