Exponents and Logarithms
Solve each problem independently before checking the answer key.
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Choose the best answer. Correct: +4, Wrong: −1, Blank: 0.
A. Radicals and Powers
The expression \(\dfrac{3x^{-1}-y^{-2}}{x^{-2}+2y^{-1}}\) without negative exponents is ....
\(\left(\dfrac{a^{\frac{2}{3}}}{b^{\frac{1}{2}}}\right)^{-1} \cdot \left(\dfrac{a^{\frac{2}{3}}b^{\frac{1}{2}}}{a^{\frac{1}{3}}}\right)^{\frac{1}{2}} = \ldots\)
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 ....
If \(x = 25\) and \(y = 64\), then \(\dfrac{x^{-\frac{2}{3}}y^{-\frac{1}{3}}}{y^{-\frac{1}{3}}x^{-\frac{1}{2}}} = \ldots\)
\(\left(\dfrac{1}{1+p}\right)^5 \left(\dfrac{1}{1-p}\right)^{-7} \left(\dfrac{p-1}{1+p}\right)^{-6} = \ldots\)
\((a - b)^{-3} \left(\dfrac{a+b}{b-a}\right)^{-2} \dfrac{1}{(a+b)^{-3}} = \ldots\)
If \(a \neq 0\), then \(\dfrac{(-2a)^3(2a)^{\frac{2}{3}}}{(16a^4)^{\frac{1}{3}}} = \ldots\)
The value of \((\sqrt{2} + \sqrt{3} + 2 + \sqrt{5})(-\sqrt{2} + \sqrt{3} + 2 - \sqrt{5})(\sqrt{10} + 2\sqrt{3}) = \ldots\)
If \(a^2 = b^{-2}c^4\), then \(c\) expressed in terms of \(a\) and \(b\) is ....
If \(n\) is an integer, then \(\dfrac{2^{n+2} \cdot 6^{n-4}}{12^{n-1}} = \ldots\)
If \(x = 4\) and \(y = 9\), then \(y\sqrt{y} - x^2\sqrt{x} + 25 = \ldots\)
\(\dfrac{(9+\sqrt{5})(2\sqrt{5}+1)}{\sqrt{5}+1} = \ldots\)
If \(p = (3 + 2\sqrt{2})^{-1}\) and \(q = (3 - 2\sqrt{2})^{-1}\), then \((1 + p)^{-1} + (1 - q)^{-1} = \ldots\)
The simplified form of \(\sqrt{7 - \sqrt{48}}\) is ....
The simplified form of \(\dfrac{(x^{-4}y^3)^{-\frac{1}{2}}(x^{-\frac{7}{3}}y^{-1})^{\frac{1}{2}}}{(x^{\frac{2}{3}}y^3)^{-\frac{1}{6}}(x^{-\frac{4}{3}}y^{-1})^3} = \ldots\)
\(\dfrac{3}{2+\sqrt{5}} + \dfrac{3}{2-\sqrt{5}} = \ldots\)
In rational exponent form, \(\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x}}} = \ldots\)
In positive exponent form, \(\dfrac{x^{-2} - y^{-2}}{(xy)^{-2}} = \ldots\)
If \(f(x) = 2^{2x} + 2^{x+1} - 3\) and \(g(x) = 2^x + 3\), then \(\dfrac{f(x)}{g(x)} = \ldots\)
Express in positive exponent and radical form: \(\dfrac{x^{-1} - y^{-1}}{x^2 + y^2} = \ldots\)
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 \(\dfrac{p}{q} = \ldots\)
If integers \(a\) and \(b\) satisfy \(\sqrt{2+\sqrt{3}} = a\sqrt{6} + b\), then \(a + b = \ldots\)
Rationalized form of \(\dfrac{1 + \frac{1}{1+\frac{1}{1}}}{1} = \ldots\)
B. Exponential Equations
Given \(f(x) = 2^{5-x} + 2^x - 12\). If \(f(x_1) = f(x_2) = 0\), then \(x_1 \cdot x_2 = \ldots\)
The value of \(x\) satisfying \(\left(\dfrac{1}{4}\right)^{x-1} = \sqrt[3]{2^{3x+1}}\) is ....
The values of \(x\) satisfying \(1000^{(x^2-3x-4)} = 10^{(x^2-2x-3)}\) are ....
If \(\sqrt[3]{8^{x+2}} = \left(\dfrac{1}{32}\right)^{(2-x)}\), then \(8x - x^2 = \ldots\)
\(2 \cdot 4^x + 2^{3-2x} = 17\), then \(2^{2x} = \ldots\)
The solution of \(2(25)^{x+1} + 5^{x+2} - 3 = 0\) is ....
The sum of the roots of \(5^{x+1} + 5^{1-x} = 11\) is ....
If \(x_1\) and \(x_2\) are roots of \(2 \cdot 9^{2x-1} - 5 \cdot 3^{2x} + 18 = 0\), then \(x_1 + x_2 = \ldots\)
The value of \(x\) satisfying \(8^{x+1} = 24^{x-1}\) is ....
The value of \(x\) satisfying \(3^{2x+3} = \sqrt{2^{7x+5}}\) is ....
The value of \(x\) satisfying \(\left(\dfrac{1}{25}\right)^{x-2,5} = \dfrac{\sqrt{625}}{\sqrt{5^{2-x}}}\) is ....
The value of \(x\) satisfying \((\sqrt{2})^{6x-4} = \left(\dfrac{1}{4}\right)^{x-9}\) is ....
The value of \(x\) satisfying \(\dfrac{0,009^{2(x-3)}}{0,3^{3x+1}} = 1\) is ....
The value of \(x\) satisfying \(\dfrac{27}{3^{2x-1}} = 81^{-0,125}\) is ....
The solution of \(\dfrac{1}{3^{-2x+2}} = 81\) is ....
The solution of \(2^{2x+2} = \dfrac{1}{8^{x+1}}\) is ....
The solution of \(\sqrt[3]{625^{2x+3}} = 5^{2x+2}\) is ....
The solution of \(3^{2x+1} + 27 = 82 \cdot 3^x\) is ....
The value of \(x\) satisfying \((\sqrt{2})^{2x-2} = 2 + \dfrac{3}{2^{-1}}\) is ....
If \(x\) satisfies \(3x^{0,4} - 9\left(\dfrac{1}{3}\right)^{0,6} = 0\), then \(3x - x^2\) equals ....
The value of \(x\) satisfying \(\dfrac{\sqrt[3]{1}}{\sqrt[3]{27}} = 3^{x+1}\) is ....
The value of \(x\) satisfying \(\dfrac{\sqrt[3]{(0,008)^{7-2x}}}{(0,2)^{-4x+5}} = 1\) is ....
If \(x = 2 - \sqrt{3}\), then \(3^{x^2-4x} = \ldots\)
The value of \(x\) satisfying \(\sqrt[3]{4^{5-x}} = \dfrac{8}{2^{2x+1}}\) is ....
If \(\dfrac{1}{2}\dfrac{\sqrt{5}}{1+\sqrt{5}} = a + b\sqrt{5}\), then \(a + b = \ldots\)
If \(4^{m+1} + 4^m = 15\), then \(8^m = \ldots\)
If \(8^m = 27\), then \(2 \cdot 4^m - 2^{m+1} = \ldots\)
If \(p^{2+{}^4\log 2} = \dfrac{{}^3\log 5}{{}^2\log 5 \cdot {}^3\log 8}\), with \(p > 0\), then \(p + p^{{}^2\log 16} = \ldots\)
If \(A^{2x} = 2\), then \(\dfrac{A^{5x} - A^{-5x}}{A^{3x} + A^{-3x}} = \ldots\)
The value of \(k\) satisfying \(x^a(x^{a+1})^a(x^a)^{1-a} = x^{k-1}\) is ....
C. Exponential Systems
The sum of values of \(x\) satisfying \(3^{4x+y} = \dfrac{1}{243}\) and \(x^2 + 7y = 25\) is ....
For \(x\) and \(y\) satisfying \(5^{x-2y+1} = 25^{x-2y}\) and \(4^{x-y+2} = 32^{x-2y+1}\), the value of \(x \cdot y = \ldots\)
The value of \(x\) satisfying \(\begin{cases} 2^{3x-2y} = \dfrac{1}{128} \\ x + 2y = 3 \end{cases}\) is ....
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 \(x + y = \ldots\)
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 \(y - x = \ldots\)
D. Exponential Inequalities
If \((2x)^{1+\frac{2}{3}\log 2x} > 64x^2\), then ....
The values of \(x\) satisfying \(x^{\sqrt{x}} > (\sqrt{x})^x\) are ....
The values of \(x\) satisfying \(4^{(x^2-x-2)} \cdot 2^{(x^2+3x-10)} < \dfrac{1}{16}\) are ....
The values of \(x\) satisfying \((3\sqrt{2})^x = 2x^2(3\sqrt{2})^{-10}\) are ....
The solution of \(\left(\dfrac{5}{x-3}\right)^2 = \dfrac{3}{125}\) is ....