Basic Mathematics - Work Sheet 2: Exponents and Radicals

Vocabulary-style flashcards covering key simplification rules, radical equations, rational exponents, and denominator rationalization from Chapter 2 of Basic Mathematics (FPMB0001).

Simplification of $y^4 \times y^2$

$$y^{4+2} = y^6$$

Simplification of $y^9 \times y^{-2}$ (Quotient $y^9 \div y^2$)

$$y^{9-2} = y^7$$

Simplification of $x^{-4} \times (x^2)^2$

$$x^{-4} \times x^4 = x^0 = 1$$

Simplification of $x^3 \div x^5$

$$x^{3-5} = x^{-2} = \frac{1}{x^2}$$

Simplification of $(3x^2)(5x)(2x^{-6})$

$$30x^{2+1-6} = 30x^{-3} = \frac{30}{x^3}$$

Simplification of $\frac{2x^3 y^5}{6x^4 y^4}$

$$\frac{y^{5-4}}{3x^{4-3}} = \frac{y}{3x}$$

Simplification of $\frac{(6x^3)^2}{(2x^2)^3 (3x^2)^0}$

$$\frac{36x^6}{8x^6 \times 1} = \frac{36}{8} = \frac{9}{2}$$

Simplification of $(2x^{\frac{3}{4}} y)^4$

$$2^4 x^3 y^4 = 16x^3 y^4$$

Simplification of $\frac{16^{\frac{3}{4}}}{32^{\frac{2}{5}}}$

$$\frac{(2^4)^{\frac{3}{4}}}{(2^5)^{\frac{2}{5}}} = \frac{2^3}{2^2} = \frac{8}{4} = 2$$

Simplification of $(-27)^{\frac{4}{3}}$

$$((-27)^{\frac{1}{3}})^4 = (-3)^4 = 81$$

Simplification of $\left(\frac{16}{81}\right)^{-\frac{3}{4}}$

$$\left(\frac{81}{16}\right)^{\frac{3}{4}} = \frac{3^3}{2^3} = \frac{27}{8}$$

Solution to $x^{\frac{2}{3}} - 5 = 4$

$$x^{\frac{2}{3}} = 9 \implies x = 9^{\frac{3}{2}} = 27$$

Solution to $(8x+5)^{\frac{1}{3}} + 2 = -1$

$$(8x+5)^{\frac{1}{3}} = -3 \implies 8x+5 = -27 \implies x = -4$$

Solution to $(3x-2)^{\frac{1}{4}} + 3 = 5$

$$(3x-2)^{\frac{1}{4}} = 2 \implies 3x-2 = 16 \implies x = 6$$

Simplification of $\sqrt{p^8}$

$$p^{\frac{8}{2}} = p^4$$

Simplification of $\sqrt[4]{16a^{12}}$

$$(16a^{12})^{\frac{1}{4}} = 2a^3$$

Simplification of $(\sqrt{3}-\sqrt{4})(\sqrt{3}+\sqrt{4})$

$$(\sqrt{3})^2 - (\sqrt{4})^2 = 3 - 4 = -1$$

Simplification of $\sqrt{32x^3} + \sqrt{50x^3}$

$$4x\sqrt{2x} + 5x\sqrt{2x} = 9x\sqrt{2x}$$

Radical form of $(2x^3 y^2)^{\frac{1}{3}}$

$$\sqrt[3]{2x^3 y^2}$$

Rational exponent form of $\sqrt[11]{x^{10}}$

$$x^{\frac{10}{11}}$$

Rationalized form of $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

$$\frac{5+3-2\sqrt{15}}{5-3} = \frac{8-2\sqrt{15}}{2} = 4-\sqrt{15}$$

Rationalized form of $\frac{\sqrt{3}}{\sqrt{5}}$

$$\frac{\sqrt{3}\times\sqrt{5}}{5} = \frac{\sqrt{15}}{5}$$