Exponents and Logarithms

Practice Questions — Radicals, Powers, Exponential Equations & Inequalities
Solve each problem independently before checking the answer key.

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

A. Radicals and Powers

1

The expression \(\dfrac{3x^{-1}-y^{-2}}{x^{-2}+2y^{-1}}\) without negative exponents is ....

A. \(\dfrac{x(3y-x)}{y(3y+2x)}\) B. \(\dfrac{x(3y^2-x)}{y(3y+2x)}\) C. \(\dfrac{x(3y^2-x)}{y(3y+x)}\) D. \(\dfrac{x(3y^2+x)}{y(3y+2x)}\) E. \(\dfrac{x(3y-x)}{y(3y+x)}\)

2

\(\left(\dfrac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}}\right)^{-1} \cdot \left(\dfrac{a^{\frac{2}{3}}b^{\frac{1}{2}}}{a^{\frac{1}{3}}}\right)^{\frac{1}{2}} = \ldots\)

A. \(\sqrt{ab}\) B. \(a\sqrt{b}\) C. \(ab\) D. \(a^{\frac{1}{3}}b^{\frac{1}{2}}\) E. \(a^{\frac{1}{3}}b^{\frac{1}{2}}\)

3

The simplified form of \(\left(\dfrac{x^{\frac{2}{3}}y^{-\frac{4}{3}}}{y^{\frac{2}{3}}x^2}\right)^{\frac{3}{4}}\) is ....

A. \(\sqrt{xy^2}\) B. \(x\sqrt{y}\) C. \(\sqrt{x^2y}\) D. \(\sqrt{xy}\) E. \(y\sqrt{x}\)

4

If \(x = 25\) and \(y = 64\), then \(\dfrac{x^{-\frac{2}{3}}y^{-\frac{1}{3}}}{y^{-\frac{1}{3}}x^{-\frac{1}{2}}} = \ldots\)

A. \(-2000\) B. \(-16\) C. \(\dfrac{16}{125}\) D. \(\dfrac{1}{16}\) E. \(2000\)

5

\(\left(\dfrac{1}{1+p}\right)^5 \left(\dfrac{1}{1-p}\right)^{-7} \left(\dfrac{p-1}{1+p}\right)^{-6} = \ldots\)

A. \(p\) B. \(1 - p^2\) C. \(p^2 + 2p + 1\) D. \(p^2 - 2p + 1\) E. \(1 + p^2\)

6

\((a - b)^{-3} \left(\dfrac{a+b}{b-a}\right)^{-2} \dfrac{1}{(a+b)^{-3}} = \ldots\)

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

7

If \(a \neq 0\), then \(\dfrac{(-2a)^3(2a)^{\frac{2}{3}}}{(16a^4)^{\frac{1}{3}}} = \ldots\)

A. \(-2^2 a\) B. \(-2a\) C. \(-2a^2\) D. \(2a\) E. \(2a^2\)

8

The value of \((\sqrt{2} + \sqrt{3} + 2 + \sqrt{5})(-\sqrt{2} + \sqrt{3} + 2 - \sqrt{5})(\sqrt{10} + 2\sqrt{3}) = \ldots\)

A. \(-4\) B. \(-2\) C. \(0\) D. \(2\) E. \(4\)

9

If \(a^2 = b^{-2}c^4\), then \(c\) expressed in terms of \(a\) and \(b\) is ....

A. \(a^{\frac{4}{3}}b^{\frac{2}{3}}\) B. \(a^{\frac{2}{3}}b^{\frac{4}{3}}\) C. \(a^{\frac{1}{2}}b^{\frac{1}{2}}\) D. \(a^2b^2\) E. \(ab\)

10

If \(n\) is an integer, then \(\dfrac{2^{n+2} \cdot 6^{n-4}}{12^{n-1}} = \ldots\)

A. \(\dfrac{1}{27}\) B. \(\dfrac{1}{16}\) C. \(\dfrac{1}{9}\) D. \(\dfrac{1}{4}\) E. \(\dfrac{1}{3}\)

11

If \(x = 4\) and \(y = 9\), then \(y\sqrt{y} - x^2\sqrt{x} + 25 = \ldots\)

A. \(-10\) B. \(-6\) C. \(6\) D. \(10\) E. \(16\)

12

\(\dfrac{(9+\sqrt{5})(2\sqrt{5}+1)}{\sqrt{5}+1} = \ldots\)

A. \(21\sqrt{5}\) B. \(19\) C. \(8\sqrt{5}\) D. \(19\sqrt{5}\) E. \(21\)

13

If \(p = (3 + 2\sqrt{2})^{-1}\) and \(q = (3 - 2\sqrt{2})^{-1}\), then \((1 + p)^{-1} + (1 - q)^{-1} = \ldots\)

A. \(1\) B. \(2\) C. \(4\) D. \(-1\) E. \(0\)

14

The simplified form of \(\sqrt{7 - \sqrt{48}}\) is ....

A. \(\sqrt{8} + \sqrt{7}\) B. \(\sqrt{7} + \sqrt{6}\) C. \(\sqrt{6} + 1\) D. \(\sqrt{5} + \sqrt{2}\) E. \(\sqrt{4} + \sqrt{3}\)

15

The simplified form of \(\dfrac{(x^{-4}y^3)^{-\frac{1}{2}}(x^{-\frac{7}{3}}y^{-1})^{\frac{1}{2}}}{(x^{\frac{2}{3}}y^3)^{-\frac{1}{6}}(x^{-\frac{4}{3}}y^{-1})^3} = \ldots\)

A. \(y\) B. \(x\) C. \(xy\) D. \(x^2y\) E. \(xy^2\)

16

\(\dfrac{3}{2+\sqrt{5}} + \dfrac{3}{2-\sqrt{5}} = \ldots\)

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

17

In rational exponent form, \(\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x}}} = \ldots\)

A. \(x^{\frac{1}{30}}\) B. \(x^{\frac{3}{10}}\) C. \(x^{\frac{1}{10}}\) D. \(x^{\frac{1}{3}}\) E. \(x^{\frac{3}{5}}\)

18

In positive exponent form, \(\dfrac{x^{-2} - y^{-2}}{(xy)^{-2}} = \ldots\)

A. \((x+y)(x-y)\) B. \(-(x+y)(x-y)\) C. \((x-y)^2\) D. \(x(x-y)\) E. \(-x(x-y)\)

19

If \(f(x) = 2^{2x} + 2^{x+1} - 3\) and \(g(x) = 2^x + 3\), then \(\dfrac{f(x)}{g(x)} = \ldots\)

A. \(2^x + 3\) B. \(2^x - 1\) C. \(2^x + 1\) D. \(2^x\) E. \(2^x - 3\)

20

Express in positive exponent and radical form: \(\dfrac{x^{-1} - y^{-1}}{x^2 + y^2} = \ldots\)

A. \(\dfrac{\sqrt{x} - \sqrt{y}}{xy}\) B. \(\dfrac{\sqrt{y} - \sqrt{x}}{xy}\) C. \(\dfrac{\sqrt{x} + \sqrt{y}}{xy}\) D. \(\dfrac{\sqrt{x} + \sqrt{y}}{x^2y^2}\) E. \(\dfrac{\sqrt{x} - \sqrt{y}}{x^2y^2}\)

21

If \(p = \left(\dfrac{x^{\frac{2}{3}} + x^{\frac{1}{3}}}{2}\right)\left(\dfrac{x^{\frac{2}{3}} - x^{\frac{1}{3}}}{3}\right)\) and \(q\) is the same, then \(\dfrac{p}{q} = \ldots\)

A. \(\sqrt[3]{x}\) B. \(\sqrt[3]{x^2}\) C. \(x\) D. \(x\sqrt[3]{x}\) E. \(x\sqrt[3]{x^2}\)

22

If integers \(a\) and \(b\) satisfy \(\sqrt{2+\sqrt{3}} = a\sqrt{6} + b\), then \(a + b = \ldots\)

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

23

Rationalized form of \(\dfrac{1 + \frac{1}{1+\frac{1}{1}}}{1} = \ldots\)

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

B. Exponential Equations

24

Given \(f(x) = 2^{5-x} + 2^x - 12\). If \(f(x_1) = f(x_2) = 0\), then \(x_1 \cdot x_2 = \ldots\)

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

25

The value of \(x\) satisfying \(\left(\dfrac{1}{4}\right)^{x-1} = \sqrt[3]{2^{3x+1}}\) is ....

A. \(\dfrac{2}{9}\) B. \(\dfrac{4}{9}\) C. \(\dfrac{5}{9}\) D. \(\dfrac{7}{9}\) E. \(\dfrac{4}{5}\)

26

The values of \(x\) satisfying \(1000^{(x^2-3x-4)} = 10^{(x^2-2x-3)}\) are ....

A. \(x_1 = 1; x_2 = \frac{9}{2}\) B. \(x_1 = -1; x_2 = \frac{9}{2}\) C. \(x_1 = -1; x_2 = \frac{7}{2}\) D. \(x_1 = 1; x_2 = -\frac{7}{2}\) E. \(x_1 = -\frac{1}{2}; x_2 = 9\)

27

If \(\sqrt[3]{8^{x+2}} = \left(\dfrac{1}{32}\right)^{(2-x)}\), then \(8x - x^2 = \ldots\)

A. \(8\) B. \(12\) C. \(15\) D. \(18\) E. \(33\)

28

\(2 \cdot 4^x + 2^{3-2x} = 17\), then \(2^{2x} = \ldots\)

A. \(\frac{1}{2}\) or \(8\) B. \(\frac{1}{2}\) or \(4\) C. \(1\) or \(4\) D. \(\frac{1}{2}\sqrt{2}\) or \(2\sqrt{2}\) E. \(\frac{1}{2}\) or \(\frac{1}{2}\sqrt{2}\)

29

The solution of \(2(25)^{x+1} + 5^{x+2} - 3 = 0\) is ....

A. \(1 - 2\log 5\) B. \(-1 - 5\log 2\) C. \(1 + 5\log 2\) D. \(-1 - 2\log 5\) E. \(-1 + 5\log 2\)

30

The sum of the roots of \(5^{x+1} + 5^{1-x} = 11\) is ....

A. \(6\) B. \(5\) C. \(0\) D. \(-2\) E. \(-4\)

31

If \(x_1\) and \(x_2\) are roots of \(2 \cdot 9^{2x-1} - 5 \cdot 3^{2x} + 18 = 0\), then \(x_1 + x_2 = \ldots\)

A. \(0\) B. \(2 - 3\log 2\) C. \(3\log 2\) D. \(2\) E. \(1\)

32

The value of \(x\) satisfying \(8^{x+1} = 24^{x-1}\) is ....

A. \(1 + 6 \cdot {}^2\log 3\) B. \(1 + 4 \cdot {}^2\log 3\) C. \(1 + 6 \cdot {}^3\log 2\) D. \(1 + 4 \cdot {}^3\log 2\) E. \(1 + 6 \cdot {}^5\log 2\)

33

The value of \(x\) satisfying \(3^{2x+3} = \sqrt{2^{7x+5}}\) is ....

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

34

The value of \(x\) satisfying \(\left(\dfrac{1}{25}\right)^{x-2,5} = \dfrac{\sqrt{625}}{\sqrt{5^{2-x}}}\) is ....

A. \(\dfrac{3}{5}\) B. \(\dfrac{8}{5}\) C. \(2\) D. \(\dfrac{12}{5}\) E. \(3\)

35

The value of \(x\) satisfying \((\sqrt{2})^{6x-4} = \left(\dfrac{1}{4}\right)^{x-9}\) is ....

A. \(-1\) B. \(0\) C. \(1\) D. \(2\) E. \(3\)

36

The value of \(x\) satisfying \(\dfrac{0,009^{2(x-3)}}{0,3^{3x+1}} = 1\) is ....

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

37

The value of \(x\) satisfying \(\dfrac{27}{3^{2x-1}} = 81^{-0,125}\) is ....

A. \(-\dfrac{3}{4}\) B. \(-\dfrac{7}{4}\) C. \(\dfrac{3}{4}\) D. \(-\dfrac{11}{4}\) E. \(\dfrac{9}{4}\)

38

The solution of \(\dfrac{1}{3^{-2x+2}} = 81\) is ....

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

39

The solution of \(2^{2x+2} = \dfrac{1}{8^{x+1}}\) is ....

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

40

The solution of \(\sqrt[3]{625^{2x+3}} = 5^{2x+2}\) is ....

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

41

The solution of \(3^{2x+1} + 27 = 82 \cdot 3^x\) is ....

A. \(\{-1, -3\}\) B. \(\{-2, 1\}\) C. \(\{-1, 3\}\) D. \(\{-1, 2\}\) E. \(\{1, 3\}\)

42

The value of \(x\) satisfying \((\sqrt{2})^{2x-2} = 2 + \dfrac{3}{2^{-1}}\) is ....

A. \(4\) B. \(2\) C. \(0\) D. \(-2\) E. \(-4\)

43

If \(x\) satisfies \(3x^{0,4} - 9\left(\dfrac{1}{3}\right)^{0,6} = 0\), then \(3x - x^2\) equals ....

A. \(3^{0,4}\) B. \(3^{0,5}\) C. \(3^{-0,25}\) D. \(3^{0,25}\) E. \(3^{-0,5}\)

44

The value of \(x\) satisfying \(\dfrac{\sqrt[3]{1}}{\sqrt[3]{27}} = 3^{x+1}\) is ....

A. \(-16\) B. \(-7\) C. \(4\) D. \(7\) E. \(16\)

45

The value of \(x\) satisfying \(\dfrac{\sqrt[3]{(0,008)^{7-2x}}}{(0,2)^{-4x+5}} = 1\) is ....

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

46

If \(x = 2 - \sqrt{3}\), then \(3^{x^2-4x} = \ldots\)

A. \(0\) B. \(\dfrac{1}{3}\) C. \(3\) D. \(9\) E. \(27\)

47

The value of \(x\) satisfying \(\sqrt[3]{4^{5-x}} = \dfrac{8}{2^{2x+1}}\) is ....

A. \(-4\) B. \(-1\) C. \(-\dfrac{1}{2}\) D. \(1\) E. \(2\)

48

If \(\dfrac{1}{2}\dfrac{\sqrt{5}}{1+\sqrt{5}} = a + b\sqrt{5}\), then \(a + b = \ldots\)

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

49

If \(4^{m+1} + 4^m = 15\), then \(8^m = \ldots\)

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

50

If \(8^m = 27\), then \(2 \cdot 4^m - 2^{m+1} = \ldots\)

A. \(12\) B. \(15\) C. \(18\) D. \(21\) E. \(24\)

51

If \(p^{2+{}^4\log 2} = \dfrac{{}^3\log 5}{{}^2\log 5 \cdot {}^3\log 8}\), with \(p > 0\), then \(p + p^{{}^2\log 16} = \ldots\)

A. \(0\) B. \(1\) C. \(2\) D. \(3\) E. \(4\)

52

If \(A^{2x} = 2\), then \(\dfrac{A^{5x} - A^{-5x}}{A^{3x} + A^{-3x}} = \ldots\)

A. \(\dfrac{31}{18}\) B. \(\dfrac{31}{9}\) C. \(\dfrac{32}{18}\) D. \(\dfrac{32}{9}\) E. \(\dfrac{33}{9}\)

53

The value of \(k\) satisfying \(x^a(x^{a+1})^a(x^a)^{1-a} = x^{k-1}\) is ....

A. \(a\) B. \(3a\) C. \(2a+1\) D. \(3a+1\) E. \(2a-1\)

C. Exponential Systems

54

The sum of values of \(x\) satisfying \(3^{4x+y} = \dfrac{1}{243}\) and \(x^2 + 7y = 25\) is ....

A. \(-23\) B. \(-17\) C. \(28\) D. \(15\) E. \(1\)

55

For \(x\) and \(y\) satisfying \(5^{x-2y+1} = 25^{x-2y}\) and \(4^{x-y+2} = 32^{x-2y+1}\), the value of \(x \cdot y = \ldots\)

A. \(6\) B. \(8\) C. \(10\) D. \(15\) E. \(20\)

56

The value of \(x\) satisfying \(\begin{cases} 2^{3x-2y} = \dfrac{1}{128} \\ x + 2y = 3 \end{cases}\) is ....

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

57

If \(x\) and \(y\) satisfy the system:
\(\begin{cases} 2^{x+1} - 3^y = 7 \\ -2^{x-1} + 3^{y+1} = 1 \end{cases}\)
then \(x + y = \ldots\)

A. \(0\) B. \(2\) C. \(3\) D. \(4\) E. \(5\)

58

If \(x\) and \(y\) satisfy the system:
\(\begin{cases} 2^{4x+7y-7} = 4^{x+3y} \\ 27^{x-y} = 3^{2x-7} \end{cases}\)
then \(y - x = \ldots\)

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

D. Exponential Inequalities

59

If \((2x)^{1+\frac{2}{3}\log 2x} > 64x^2\), then ....

A. \(\dfrac{1}{4} < x < 4\) B. \(x < \dfrac{1}{4}\) or \(x > 4\) C. \(0 < x < \dfrac{1}{4}\) or \(x > 4\) D. \(0 < x < \dfrac{1}{4}\) or \(x > 2\) E. \(x > \dfrac{1}{4}\)

60

The values of \(x\) satisfying \(x^{\sqrt{x}} > (\sqrt{x})^x\) are ....

A. \(0 < x < 1\) or \(2 < x < 4\) B. \(x \leq 2\) C. \(1 < x < 4\) D. \(2 \leq x \leq 3\) E. \(1 < x < 6\)

61

The values of \(x\) satisfying \(4^{(x^2-x-2)} \cdot 2^{(x^2+3x-10)} < \dfrac{1}{16}\) are ....

A. \(x < -5\) or \(x > -2\) B. \(x < -2\) or \(x > \dfrac{5}{3}\) C. \(-2 < x < -1\) D. \(-2 < x < \dfrac{5}{3}\) E. \(-5 < x < 2\)

62

The values of \(x\) satisfying \((3\sqrt{2})^x = 2x^2(3\sqrt{2})^{-10}\) are ....

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

63

The solution of \(\left(\dfrac{5}{x-3}\right)^2 = \dfrac{3}{125}\) is ....

A. \(2\dfrac{1}{2}\) B. \(3\dfrac{1}{2}\) C. \(4\dfrac{1}{2}\) D. \(5\dfrac{1}{2}\) E. \(6\dfrac{1}{2}\)

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