Problems – Exponents

EXPONENTS — Practice Questions

36 multiple-choice questions on exponents and radicals.
Try to solve each question independently before checking the answer key.

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Instructions:
Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.

Quick Notes:
Exponent Rules:
\(a^m \cdot a^n = a^{m+n}\)
\(\dfrac{a^m}{a^n} = a^{m-n}\)
\((a^m)^n = a^{mn}\)
\((ab)^n = a^n b^n\)
\(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\)
\(a^{-n} = \dfrac{1}{a^n}\)
\(a^0 = 1\) (for \(a \neq 0\))

Radical Rules:
\(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\)
\(\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}\)

Problem 1

Simplify form from \(\frac{1}{1+x^{p-q}} + \frac{1}{1+x^{q-p}}\) is….

A. \(0\) B. \(1\) C. \(2\) D. \(x^{p-q}\) E. \(x^{q-p}\)

Problem 2

Simplify form of \(\frac{(x+y)^{3a+1}}{(x+y)^{2a+5}}\) is….

A. \((x+y)^{a-4}\) B. \((x+y)^{5a+6}\) C. \((x+y)^{a+6}\) D. \((x+y)^{5a-4}\) E. \((x+y)^{a+4}\)

Problem 3

Simplify form of \(\left(\frac{a^2 b}{c^2}\right)^3 \times \frac{b^4}{ac^3}\) is….

A. \(\frac{a^5 b^7}{c^9}\) B. \(\frac{a^5 b^7}{c^3}\) C. \(\frac{a^7 b^7}{c^9}\) D. \(\frac{a^5 b^3}{c^9}\) E. \(\frac{a^7 b^3}{c^3}\)

Problem 4

If \(3^x + 3^{-x} = 6\), for \(9^x + 9^{-x} = \cdots\)

A. \(34\) B. \(36\) C. \(38\) D. \(40\) E. \(42\)

Problem 5

The value of \(x\) from the equation \(\left(\frac{3}{3^{x-2}}\right)^2 = \sqrt[3]{\frac{1}{9}}\) is….

A. \(1\) B. \(2\) C. \(3\) D. \(4\) E. \(5\)

Problem 6

The value of \(\dfrac{\sqrt[3]{8^{(\frac{1}{3})^9}}}{(\sqrt[3]{2})^6}\) is….

A. \(1\) B. \(2\) C. \(4\) D. \(8\) E. \(16\)

Problem 7

Simplify results of \(\sqrt[3]{\sqrt{\frac{27}{8}}} = \cdots\)

A. \(\frac{3}{2}\) B. \(\frac{3}{4}\) C. \(\frac{9}{4}\) D. \(\frac{\sqrt{3}}{\sqrt{2}}\) E. \(\frac{\sqrt{27}}{8}\)

Problem 8

If \(a>0\), for \(\left(a^{\frac{1}{2}} - a^{-\frac{1}{2}}\right)^2 \left(a^{\frac{1}{2}} + a^{-\frac{1}{2}}\right)^2\) same as….

A. \(a^2 - a^{-2}\) B. \(a^2 + a^{-2}\) C. \(a - a^{-1}\) D. \(a + a^{-1}\) E. \(a^2 + a^{-2} - 2\)

Problem 9

If \(p=1+\sqrt{3}\), for \(p^2-2\) is….

A. \(2+\sqrt{3}\) B. \(2+2\sqrt{3}\) C. \(4+2\sqrt{3}\) D. \(4+\sqrt{3}\) E. \(2\)

Problem 10

Result of \(\left(-z^5 u^5\right)^3 = \cdots\)

A. \(-z^{15} u^{15}\) B. \(z^{15} u^{15}\) C. \(-z^8 u^8\) D. \(z^8 u^8\) E. \(-z^5 u^5\)

Problem 11

Simple form of \(\left(\frac{x^2}{y^3}\right)^6 : \left(\frac{y^6}{x^{-4}}\right)^{-3}\) is….

A. \(x^{24} y^{36}\) B. \(x^{24} y^0\) C. \(x^0 y^{36}\) D. \(x^{24} y^{18}\) E. \(x^{12} y^{36}\)

Problem 12

The value of \(\frac{\sqrt{0.0036}}{0.9}\) is….

A. \(\frac{1}{15}\) B. \(\frac{2}{15}\) C. \(\frac{3}{15}\) D. \(\frac{4}{15}\) E. \(\frac{5}{15}\)

Problem 13

The value of \(x\) from the equation \(8^{2x-1} = \sqrt{4^{2x+3}}\) is….

A. \(\frac{1}{3}\) B. \(\frac{2}{3}\) C. \(\frac{3}{2}\) D. \(\frac{1}{5}\) E. \(\frac{2}{5}\)

Problem 14

If \(\sqrt{3\sqrt{9\sqrt{27}}} = 3^{\frac{x}{y}}\), then the value \(x+y=\cdots\)

A. \(15\) B. \(17\) C. \(19\) D. \(21\) E. \(23\)

Problem 15

Result of \(\frac{5^{2-n} - (0.2)^n}{5^{1-n} + (0.2)^n}\) is….

A. \(\frac{24}{7}\) B. \(\frac{25}{6}\) C. \(\frac{24}{5}\) D. \(\frac{25}{7}\) E. \(\frac{24}{6}\)

Problem 16

The solution set from \(5^{2x+1} - 6 \cdot 5^x + 1 = 0\) is….

A. \(\{0,1\}\) B. \(\{-1,0\}\) C. \(\{1,2\}\) D. \(\{-2,-1\}\) E. \(\{0,2\}\)

Problem 17

Set of solutions to equations \(9^{3x} - 2 \cdot 3^{3x+1} - 27 = 0\) is….

A. \(\{1\}\) B. \(\{2\}\) C. \(\{3\}\) D. \(\{4\}\) E. \(\{5\}\)

Problem 18

Shape \(\left(\frac{6}{5}\right)^3\) can be written as….

A. \(\frac{216}{125}\) B. \(\frac{125}{216}\) C. \(\frac{36}{25}\) D. \(\frac{25}{36}\) E. \(\frac{216}{25}\)

Problem 19

Result of \(\frac{3\sqrt{5} \times 5\sqrt{5}}{25}\) is….

A. \(1\) B. \(2\) C. \(3\) D. \(4\) E. \(5\)

Problem 20

The value of \(\frac{\sqrt{10} - \sqrt{5}}{\sqrt{5}}\) is….

A. \(\sqrt{2} - 1\) B. \(\sqrt{2} + 1\) C. \(\sqrt{5} - 1\) D. \(\sqrt{5} + 1\) E. \(1 - \sqrt{2}\)

Problem 21

The rational form of \(\frac{5}{3\sqrt{2} - \sqrt{3}}\) is….

A. \(\frac{15\sqrt{2}+5\sqrt{3}}{15}\) B. \(\frac{15\sqrt{2}+5\sqrt{3}}{3}\) C. \(\frac{15\sqrt{2}+5\sqrt{3}}{21}\) D. \(\frac{15\sqrt{2}+5\sqrt{3}}{25}\) E. \(\frac{15\sqrt{2}+5\sqrt{3}}{30}\)

Problem 22

The results of the operation from \(\sqrt{18} + \sqrt{50} - \sqrt{72}\) is….

A. \(2\sqrt{2}\) B. \(3\sqrt{2}\) C. \(4\sqrt{2}\) D. \(5\sqrt{2}\) E. \(6\sqrt{2}\)

Problem 23

The value of \(\sqrt{31 + \sqrt{936}} - \sqrt{21 - \sqrt{416}}\) is….

A. \(4\) B. \(5\) C. \(6\) D. \(7\) E. \(8\)

Problem 24

It is known \(\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} = a + b\sqrt{6}\); If \(a\) and \(b\) are integers, then \(a+b=\cdots\)

A. \(-5\) B. \(-4\) C. \(-3\) D. \(-2\) E. \(-1\)

Problem 25

Simple form of \(\sqrt[m]{\sqrt[n]{a^p}} = \cdots\)

A. \(a^{\frac{p}{mn}}\) B. \(a^{\frac{mn}{p}}\) C. \(a^{\frac{p}{m+n}}\) D. \(a^{\frac{m+n}{p}}\) E. \(a^{\frac{m}{pn}}\)

Problem 26

If \(m = \sqrt{18} + \sqrt{80}\), then the value \(\left(m - 7\sqrt{2}\right)\left(m + 7\sqrt{2}\right)\) is….

A. \(20\) B. \(30\) C. \(40\) D. \(50\) E. \(60\)

Problem 27

Simplified form of \(2\sqrt{8} + \sqrt{18} + \frac{1}{4}\sqrt{32} + \sqrt{200}\) is….

A. \(\frac{51}{2}\sqrt{2}\) B. \(\frac{53}{2}\sqrt{2}\) C. \(\frac{55}{2}\sqrt{2}\) D. \(\frac{57}{2}\sqrt{2}\) E. \(\frac{59}{2}\sqrt{2}\)

Problem 28

If \(\sqrt[3]{49^3\sqrt[3]{49^3} \sqrt[3]{49^3}\sqrt[3]{\cdots}} = a\), then the value \(a\) is….

A. \(49\) B. \(98\) C. \(147\) D. \(196\) E. \(245\)

Problem 29

Simplified form of \(\frac{3\sqrt{24} - 2\sqrt{18}}{-\sqrt{2}}\) is….

A. \(-6\) B. \(-5\) C. \(-4\) D. \(-3\) E. \(-2\)

Problem 30

A right triangle has a base length of \((5 + \sqrt{2})\) cm and height \((5 - \sqrt{2})\) cm. So the perimeter of the triangle is….

A. \(10 + 2\sqrt{23}\) B. \(10 + 3\sqrt{23}\) C. \(10 + 4\sqrt{23}\) D. \(10 + 5\sqrt{23}\) E. \(10 + 6\sqrt{23}\)

Problem 31

It is known that the radius of a ball is \(\left(2\sqrt{2} + \sqrt{6}\right)\) cm. So the surface area of the ball is .... \(cm^2\).

A. \(8\pi(7 + 4\sqrt{3})\) B. \(8\pi(7 + 2\sqrt{3})\) C. \(4\pi(7 + 4\sqrt{3})\) D. \(4\pi(7 + 2\sqrt{3})\) E. \(2\pi(7 + 4\sqrt{3})\)

Problem 32

If \(\left(\frac{3}{3^{x-2}}\right)^2 = \sqrt[3]{\frac{1}{9}}\), for \(x = \cdots\)

A. \(\frac{5}{3}\) B. \(\frac{7}{3}\) C. \(\frac{10}{3}\) D. \(\frac{11}{3}\) E. \(\frac{13}{3}\)

Problem 33

Mark \(x\) which satisfies the equation \(3^{x+3} = \sqrt[5]{27^{x-5}}\) is….

A. \(5\) B. \(7\) C. \(9\) D. \(11\) E. \(13\)

Problem 34

The value of \(x\) that satisfies the equation \((0.25)^{x+4} = \sqrt{8^{2x-5}}\) is….

A. \(1\) B. \(2\) C. \(3\) D. \(4\) E. \(5\)

Problem 35

Simple results of \(\sqrt{108} - \frac{2}{3-\sqrt{27}} = \cdots\)

A. \(4\sqrt{3} - 4\) B. \(6\sqrt{3} - 6\) C. \(8\sqrt{3} - 8\) D. \(10\sqrt{3} - 10\) E. \(12\sqrt{3} - 12\)

Problem 36

Simple form of \(\frac{(3p^{-2} q^3)^{-2}}{9p^{-1}q^2} = \cdots\)

A. \(\frac{p^5}{q^8}\) B. \(\frac{p^4}{81q^8}\) C. \(\frac{p^5}{81q^8}\) D. \(\frac{p^5}{q^8}\) E. \(\frac{p^5}{q^4}\)