If \(5^{a}=3125\), then the value of \(5^{(a-3)}\) is:
Answer: A
\(5^{a}=3125 \Leftrightarrow 5^{a}=5^{5} \)
\(\Rightarrow a = 5 \)
\(\therefore 5^{(a-3)}=5^{(5-3)}=5^{2}=25\)
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\((0.04)^{-1.5}\) = ?
Answer: B
\((0.04)^{-1.5} = (\frac{4}{100})^{-1.5} \)
\(= (\frac{1}{25})^{-\frac{3}{2}} \)
\(= (25)^{\frac{3}{2}} \)
\(= (5^{2})^{\frac{3}{2}} \)
\(= (5)^{2\times \frac{3}{2}} \)
\(= 5^{3} = 125\)
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\(125 \times 125 \times 125 \times 125 \times 125 = 5^?\)
Answer: C
\[ 125 = 5^{3} \quad\Longrightarrow\quad 125^{5} = (5^{3})^{5} = 5^{15} \]
So the exponent is 15. Multiplying five copies of \( 5^{3} \) adds their exponents.
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\((17)^{3.5}\times(17)^{?}=17^8\)
Answer: D
Let \((17)^{3.5}\times(17)^x=17^8\)
Then, \((17)^{3.5+x}=17^8\)
\(\therefore 3.5+x=8\)
\(\Rightarrow x=(8-3.5)\)
\(\Rightarrow x=4.5\)
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If \(x=3+2\sqrt{2}\), then the value of \((\sqrt{x}-\frac{1}{\sqrt{x}})\) is:
Answer: B
\((\sqrt{x}-\frac{1}{\sqrt{x}})^{2}=x+\frac{1}{x}-2\)
\(= (3+2\sqrt{2})+\frac{1}{(3+2\sqrt{2})}-2\)
\(= (3+2\sqrt{2})+\frac{1}{(3+2\sqrt{2})}\times\frac{(3-2\sqrt{2})}{(3-2\sqrt{2})}-2\)
\(= (3+2\sqrt{2})+(3+2\sqrt{2})-2\)
= 4
\(\therefore (\sqrt{x}-\frac{1}{\sqrt{x}})=2\)
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The value of \((\sqrt8)^{\frac{1}{3}}\) is:
Answer: C
\((\sqrt8)^{\frac{1}{3}}\)
\(= (8^{\frac{1}{2}})^{\frac{1}{3}}\)
\(= 8^{\frac{1}{6}}\)
\(= (2^3)^{\frac{1}{6}}\)
\(= 2^{\frac{1}{2}}\)
\(= \sqrt{2}\)
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If \(a^x = b, b^y = c\) and \(c^z = a\), then the value of \(xyz\) is:
Answer: A
Substitute each equation into the next.
\[ c = b^{y} = (a^{x})^{y} = a^{xy} \]
\[ a = c^{z} = (a^{xy})^{z} = a^{xyz} \]
Comparing exponents on the same base:
\[ xyz = 1 \]
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If \(7^a = 16807\), then the value of \(7^{(a-3)}\) is:
Answer: A
\[ 16807 = 7^{5} \quad\Longrightarrow\quad a = 5 \]
\[ 7^{a-3} = 7^{2} = 49 \]
Equivalently \( \tfrac{16807}{343} = 49 \).
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\(\frac{(243)^{n/5}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}=?\)
Answer: C
Given Expression = \(\frac{(243)^{(n/5)}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}\)
\(=\frac{(3^{5})^{(n/5)}\times 3^{2n+1}}{(3^{2})^{n}\times 3^{n-1}}\)
\(=\frac{(3^{5\times(n/5))}\times 3^{2n+1}}{(3^{2n}\times 3^{n-1})}\)
\(=\frac{3^{n\times3^{2n+1}}}{3^{2n}\times3^{n-1}}\)
\(=\frac{3^{(n+2n+1)}}{3^{(2n+n-1)}}\)
\(=\frac{3^{3n+1}}{3^{3n-1}}\)
\(=3^{(3n+1-3n+1)}=3^{2}=9\)
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Solve for \(m\) if \(49(7^m) = 343^{3m+6}\)
Answer: B
Exp: \(49(7^m) = 343^{3m+6} \)
\(\Rightarrow 7^2 7^m = (7^3)^{3m+6} \)
\(\Rightarrow 7^{2+m} = 7^{9m+18}\)
Equating powers of 7 on both sides,
\(\Rightarrow m + 2 = 9m + 18\)
\(\Rightarrow - 16 = 8m\)
\(\Rightarrow m = -2\)
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