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PSAT Math : Pattern Behaviors in Exponents

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Example Questions

Example Question #254 : Exponents

Solve for

$\left ( \frac{2}{3} \right )^{x+1} = \frac{27}{8}$

Possible Answers:

None of the above

Correct answer:

Explanation:

$\left ( \frac{2}{3} \right )^{x+1} = \frac{27}{8}$ = $\left ( \frac{3}{2} \right )^{3} = \left ( \frac{2}{3} \right )^{-3}$

x+1=-3 which means x=-4

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Example Question #255 : Exponents

Which of the following statements is the same as:

$\dfrac{4^{x+2}}{2^y}$

$I)\ 2^{2x+4-y} \quad II)\ 2^{x+2} / 4^y \quad III)\ (4^x)( 4^2 )(2^{-y} )$

Possible Answers:

$II \mbox{ and } III \mbox{ only}$

$I \mbox{ and } III \mbox{ only}$

$I, II \mbox{ and } III$

$I \mbox{ and } II \mbox{ only}$

$I \mbox{ only}$

Correct answer:

$I \mbox{ and } III \mbox{ only}$

Explanation:

Remember the laws of exponents. In particular, when the base is nonzero:

$b^{m+n} = b^m b^n \qquad (b^m)^n = b^{mn} \\ (b \times c)^n = b^n c^n \qquad b^{-n} = 1/b^n$

An effective way to compare these statements, is to convert them all into exponents with base 2. The original statement becomes:

$4^{x+2} / 2^y = 4^{x+2} (2^{-y}) = (2^2)^{x+2} 2^{-y} = 2^{2(x+2) } 2^{-y} = 2^{2x+4-y}$

This is identical to statement I. Now consider statement II:

$2^{x+2} / 4^y = 2^{x+2} ( 4^{-y} ) = 2^{x+2} ( 2^2 )^{-y} = 2^{x+2} 2^{-2y} = 2^{x+2-2y} \neq 2^{2x+4-y}$

Therefore, statement II is not identical to the original statement. Finally, consider statement III:

$(4^x)(4^2 ) (2^{-y}) = (2^2)^x (2^2)^2 (2^{-y}) = (2^{2x} )( 2^4 ) (2^{-y}) = 2^{2x+4-y}$

which is also identical to the original statement. As a result, only I and III are the same as the original statement.

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Example Question #256 : Exponents

Write in exponential form:

$\sqrt[3]{\left ( x-1 \right )^{2}}$

Possible Answers:

$\left ( x-1 \right )$

$\left ( x-1 \right )^{3}$

$\left ( x-1 \right )^{2}$

$\left ( x-1 \right )^{\frac{2}{3}}$

$\left ( x-1 \right )^{\frac{3}{2}}$

Correct answer:

$\left ( x-1 \right )^{\frac{2}{3}}$

Explanation:

Using properties of radicals e.g., $\sqrt[m]{x^{n}} = x^{\frac{n}{m}}$

we get $\sqrt[3]{\left ( x-1 \right )^{2}} = \left ( x-1 \right )^{\frac{2}{3}}$

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Example Question #5 : How To Find Patterns In Exponents

Write in exponential form:

$\sqrt[3]{x^{5}}$

Possible Answers:

$x^{\frac{3}{5}}$

$x^{3}$

$x^{5}$

$x^{5-3}$

$x^{\frac{5}{3}}$

Correct answer:

$x^{\frac{5}{3}}$

Explanation:

Properties of Radicals

$\sqrt[]{x} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{\frac{1}{3}}$

$x^{\frac{5}{3}}$

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Example Question #11 : How To Find Patterns In Exponents

Write in radical notation:

$\left ( x+3 \right )^{\frac{2}{3}}$

Possible Answers:

$\sqrt[]{\left ( x+3 \right )^{2}}$

$\sqrt[3]{\left ( x+3 \right )^{2}}$

$\sqrt[3]{\left ( x+3 \right )}$

$\sqrt[]{\left ( x+3 \right )}$

$x^{\frac{2}{3}} + 3^{\frac{2}{3}}$

Correct answer:

$\sqrt[3]{\left ( x+3 \right )^{2}}$

Explanation:

Properties of Radicals

$\sqrt[]{x} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{\frac{1}{3}}$

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Example Question #12 : How To Find Patterns In Exponents

Express in radical form : $x^{\frac{3}{5}}$

Possible Answers:

$x^{0.6}$

$\sqrt[3]{x^{5}}$

$x^{15}$

$\sqrt[5]{x^{3}}$

$\sqrt[3]{x}$

Correct answer:

$\sqrt[5]{x^{3}}$

Explanation:

Properties of Radicals

$\sqrt[]{x} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{\frac{1}{3}}$

$x^{\frac{3}{5}}=\sqrt[5]{x^{3}}$

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Example Question #11 : Pattern Behaviors In Exponents

Simplify: $\sqrt[3]{2}\star \sqrt[3]{32}$

Possible Answers:

$1.26\star 3.17$

$\sqrt[3]{\frac{32}{2}} = \sqrt[3]{16} = 2.52$

$\sqrt[3]{\frac{2}{32}} = \sqrt[3]{\frac{1}{16}} = \frac{1}{2.52}$

$8\star 96 = 768$

$\sqrt[3]{2}\star \sqrt[3]{32} = \sqrt[3]{64} = 4$

Correct answer:

$\sqrt[3]{2}\star \sqrt[3]{32} = \sqrt[3]{64} = 4$

Explanation:

$\sqrt[3]{2}\star \sqrt[3]{32}= \sqrt[3]{64}= 4$

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Example Question #265 : Exponents

Simplify:

$\frac{\sqrt[3]{x^{5}y^{-2}}}{\sqrt[3]{x^{-1}y^{4}}}$

Possible Answers:

$\frac{x^{2}}{y^{2}}$

$\frac{x^{15}y^{-6}}{x^{-3}y^{12}}$

$\frac{\sqrt[3]{x^{5}}}{\sqrt[3]{y^{4}}}$

$\frac{x\sqrt[3]{x^{2}}}{y\sqrt[3]{y}}$

$\frac{\sqrt[3]{y^{-2}}}{\sqrt[3]{x^{-1}}}$

Correct answer:

$\frac{x^{2}}{y^{2}}$

Explanation:

Convert the given expression into a single radical e.g. $\sqrt[3]{\frac{x^{5}y^{-2}}{x^{-1}y^{4}}}$ the expression inside the radical is:

$\frac{x^{6}}{y^{6}}$

and the cube root of this is :

$\frac{x^{2}}{y^{2}}$

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Example Question #261 : Exponents

Solve for .

$\log _{\frac{1}{5}}\left ( \frac{1}{25} \right )=x$

Possible Answers:

Correct answer:

Explanation:

$\frac{1}{25}= \left ( \frac{1}{5} \right )^{x}$

Hence must be equal to 2.

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Example Question #267 : Exponents

Simplify:

$log\left (\frac{x}{y^{2}} \right )$

Possible Answers:

2logy-logx

$logx+logy^{2}$

logx-2logy

$\left ( logx \right )\left ( logy^{2} \right )$

$logx-logy^{2}$

Correct answer:

logx-2logy

Explanation:

$log\left ( \frac{x}{y^{2}} \right )=logx-logy^{2}$

Now $logy^{2} = 2logy$

Hence the correct answer is logx - 2logy

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PSAT Math : Pattern Behaviors in Exponents

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #254 : Exponents

Example Question #255 : Exponents

Example Question #256 : Exponents

Example Question #5 : How To Find Patterns In Exponents

Example Question #11 : How To Find Patterns In Exponents

Example Question #12 : How To Find Patterns In Exponents

Example Question #11 : Pattern Behaviors In Exponents

Example Question #265 : Exponents

Example Question #261 : Exponents

Example Question #267 : Exponents

All PSAT Math Resources

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