\((256)^{0.16} \times (256)^{0.09} = ?\)
Answer: A
\((256)^{0.16} \times (256)^{0.09} = (256)^{(0.16+0.09)} \)
\(= (256)^{0.25} \)
\(= (256)^{(25/100)} \)
\(= (256)^{(1/4)} \)
\(= (4^{4})^{(1/4)} \)
\(= 4^{4(1/4)} \)
\(= 4^{1} \)
\(= 4\)
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If \(\sqrt{2n}=64\), then the value of \(n\) is:
Answer: D
\(\sqrt{2n}=64 \)
\(\Rightarrow 2^{\frac{n}{2}}=64=2^6 \)
\(\Rightarrow \frac{n}{2}=6 \)
\(\therefore n=12\)
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The value of \((\frac{32}{243})^{-\frac{4}{5}}\) is:
Answer: D
\((\frac{32}{243})^{-\frac{4}{5}} \)
\(= (\frac{243}{32})^{\frac{4}{5}} \)
\(= [(\frac{3}{2})^5]^{\frac{4}{5}} \)
\(= \frac{81}{16}\)
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Given that \(10^{0.48}=x\), \(10^{0.70}=y\) and \(x^{z}=y^{2}\), then the value of z is close to:
Answer: C
\(x^{z}=y^{2}\) \(\Leftrightarrow 10^{(0.48z)}=10^{2\times 0.70}=10^{1.40}\)
\(\Rightarrow 0.48z=1.40\)
\(\Rightarrow z=\frac{140}{48}=\frac{35}{12}=2.9(approx.)\)
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\((2^{n+4}-2.2^{n})\div(2.2^{n+3})=2^{-3}\) is equal to:
Answer: D
\((2^{n+4}-2.2^{n})\div2.2^{n+3}+\frac{1}{2}^{3} \)
\(= \frac{7}{8}+\frac{1}{8} \)
\(= 1\)
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If m and n are whole numbers such that \(m^{n}\) = 121, the value of \((m-n)^{n+1}\) is:
Answer: D
We know that \(11^{2}\) =121.
Putting m = 11 and n = 2, we get:
\((m-1)^{n+1} = (11-1)^{(2+1)} = 10^{3} = 1000\)
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\((\frac{x^{b}}{x^{c}})^{(b+c-a)}.(\frac{x^{c}}{x^{a}})^{(c+a-b)}.(\frac{x^{a}}{x^{b}})^{(a+b-c)}=?\)
Answer: B
Given exp. = \(x^{(b-c)(b+c-a)}.x^{(c-a)(c+a-b)}.x^{(a-b)(a+b-c)}\)
\(=x^{(b-c)(b+c)-a(b-c)}.x^{(c-a)(c+a)-b(c-a)}.x^{(a-b)(a+b)-c(a-b)}\)
\(= x^{(b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2})}\)
\(= (x^{0}xx^{0})\)
\(= (1\times1)=1\)
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If \(m\) and \(n\) are whole numbers such that \(m^n = 169\), then the value of \((m-1)^{n+1}\) is:
Answer: D
Exp: Clearly, \(m = 13\), and \(n = 2\).
Therefore, \((m-1)^{n+1} = (13 - 1)^3 = 12^3 = 1728\)
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If \(5\sqrt{5}\times5^3\div5^{-\frac{3}{2}}=5^{a+2}\), the value of \(a\) is:
Answer: A
\(5^{\frac{3}{2}}\times5^3\div5^{3\frac{3}{2}}=5^{a+2} \)
\(\Rightarrow 5^{\frac{3}{2}}+3+\frac{3}{2}=5^{a+2} \)
\(\Rightarrow \frac{3}{2}+3+\frac{3}{2}=a+2 \)
\(\Rightarrow a+2=6 \)
\(\therefore a=4\)
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If \(a\) and \(b\) are positive numbers, \(2^a = b^3\) and \(b^a = 8\), find the value of \(a\) and \(b\)
Answer: B
Exp: \(2^a = b^3 ....(1)\)
\(b^a = 8 ....(2)\)
cubing both sides of equation \((2), (b^a)^3 = 8^3\)
\(\Rightarrow b^{3a} = (b^3)^a = 512\).
from \((1), (2^a)^a = (2^3)^3\).
comparing both sides, a = 3
substiting a in (1), b = 2
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