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PSAT Math : How to find an exponent from a rational number

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Example Questions

Example Question #1 : How To Find An Exponent From A Rational Number

Com_exp_1

Which of the following lists the above quantities from least to greatest?

Possible Answers:

I, IV, III, II

II, III, I, IV

I, III, II, IV

I, IV, II, III

IV, III, II, I

Correct answer:

I, III, II, IV

Explanation:

Com_exp_2

Com_exp_3

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Example Question #2 : How To Find An Exponent From A Rational Number

Solve for $\displaystyle x$.

$2^{x}= 64$ $\displaystyle 2^{x}= 64$

Possible Answers:

x=64 $\displaystyle x=64$

$x=64^{2}$ $\displaystyle x=64^{2}$

x=6 $\displaystyle x=6$

x=-6 $\displaystyle x=-6$

x=5 $\displaystyle x=5$

Correct answer:

x=6 $\displaystyle x=6$

Explanation:

Since $2^{x}= 2^{6}$ $\displaystyle 2^{x}= 2^{6}$

Hence x=6 $\displaystyle x=6$

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Example Question #3 : How To Find An Exponent From A Rational Number

Simplify:

$\sqrt[3]{x^{10}y^{8}z^{9}}$ $\displaystyle \sqrt[3]{x^{10}y^{8}z^{9}}$

Possible Answers:

$\sqrt[3]{xy^{2}}$ $\displaystyle \sqrt[3]{xy^{2}}$

$x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$ $\displaystyle x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

$x^{3}y^{2}z^{3}^{\sqrt[3]{xyz}}$ $\displaystyle x^{3}y^{2}z^{3}^{\sqrt[3]{xyz}}$

$x^{3}y^{3}z^{3}$ $\displaystyle x^{3}y^{3}z^{3}$

$x^{3}y^{2}z^{3}$ $\displaystyle x^{3}y^{2}z^{3}$

Correct answer:

$x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$ $\displaystyle x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

Explanation:

$\sqrt[3]{x^{9}xy^{6}y^{2}z^{9}}=x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$ $\displaystyle \sqrt[3]{x^{9}xy^{6}y^{2}z^{9}}=x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

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Example Question #4 : How To Find An Exponent From A Rational Number

Solve for $\displaystyle x$:

$\sqrt{x} -1= 4$ $\displaystyle \sqrt{x} -1= 4$

Possible Answers:

$\displaystyle 36$

$\displaystyle 25$

$\displaystyle 5$

$\displaystyle 9$

$\displaystyle 5$

Correct answer:

$\displaystyle 25$

Explanation:

From the equation one can see that

$\sqrt{x} = 5$ $\displaystyle \sqrt{x} = 5$

Hence $\displaystyle x$ must be equal to 25.

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Example Question #5 : How To Find An Exponent From A Rational Number

Evaluate:

$\log 0.1=$ $\displaystyle \log 0.1=$

Possible Answers:

$\displaystyle -1$

$\displaystyle -2$

$\frac{1}{10}$ $\displaystyle \frac{1}{10}$

$\displaystyle 1$

$\displaystyle 10$

Correct answer:

$\displaystyle -1$

Explanation:

$0.1= 10^{-1}$ $\displaystyle 0.1= 10^{-1}$

$\log 0.1=-1$ $\displaystyle \log 0.1=-1$

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Example Question #6 : How To Find An Exponent From A Rational Number

Solve for $\displaystyle x$.

$\log_{2}x=4$ $\displaystyle \log_{2}x=4$

Possible Answers:

$\displaystyle 16$

$\displaystyle 32$

$\displaystyle 64$

$\displaystyle 8$

$\displaystyle 1$

Correct answer:

$\displaystyle 16$

Explanation:

$x= 2^4{}= 16$ $\displaystyle x= 2^4{}= 16$

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Example Question #7 : How To Find An Exponent From A Rational Number

$\log 3=a$ $\displaystyle \log 3=a$

and

$\log 5=b$ $\displaystyle \log 5=b$

Find $\log 15$ $\displaystyle \log 15$

Possible Answers:

$a^{2}$ $\displaystyle a^{2}$

$\displaystyle a$

$b^{2}$ $\displaystyle b^{2}$

$\displaystyle b$

a+b $\displaystyle a+b$

Correct answer:

a+b $\displaystyle a+b$

Explanation:

$\log 15=\log (5\cdot 3)=\log 5+\log 3$ $\displaystyle \log 15=\log (5\cdot 3)=\log 5+\log 3$

Hence the correct answer is a+b $\displaystyle a+b$.

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Example Question #8 : How To Find An Exponent From A Rational Number

Solve for $\displaystyle x$.

$\log (x+3)+\log x=1$ $\displaystyle \log (x+3)+\log x=1$

Possible Answers:

$-5\ or\ 2$ $\displaystyle -5\ or\ 2$

$\displaystyle 10$

$\displaystyle 5$

$5\ or\ -2$ $\displaystyle 5\ or\ -2$

$\displaystyle 2$

Correct answer:

$\displaystyle 2$

Explanation:

Use the rules of logarithms to combine terms.

$\log (x(x+3))=1$ $\displaystyle \log (x(x+3))=1$

Hence, $x(x+3)=10^{1}$ $\displaystyle x(x+3)=10^{1}$

$x^{2}+3x=10$ $\displaystyle x^{2}+3x=10$

$\displaystyle x^2+3x-10=0$

By fatoring we get $\displaystyle (x+5)(x-2)=0$

Hence $x=-5\ or\ 2$ $\displaystyle x=-5\ or\ 2$.

However, you cannot take the logarithm of a negative number. Thus, the only value for $\displaystyle x$ is $\displaystyle 2$.

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Example Question #9 : How To Find An Exponent From A Rational Number

If a piece of pie is cut into 3 sections, and each of those pieces is further cut into three sections, then those pieces are cut into three sections, how many (tiny) pieces of pie are there?

Possible Answers:

125

Correct answer:

Explanation:

The answer is 3³ = 27

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Example Question #10 : How To Find An Exponent From A Rational Number

Solve for $\displaystyle x$.

$\log _{9}x=\frac{1}{2}$ $\displaystyle \log _{9}x=\frac{1}{2}$

Possible Answers:

$\displaystyle 9$

$\displaystyle 81$

$\displaystyle 3$

$\displaystyle 1$

$\displaystyle 27$

Correct answer:

$\displaystyle 3$

Explanation:

$\log _{9}x=\frac{1}{2}$ $\displaystyle \log _{9}x=\frac{1}{2}$

$x = 9^{\frac{1}{2}}= 3$ $\displaystyle x = 9^{\frac{1}{2}}= 3$

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PSAT Math : How to find an exponent from a rational number

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Find An Exponent From A Rational Number

Example Question #2 : How To Find An Exponent From A Rational Number

Example Question #3 : How To Find An Exponent From A Rational Number

Example Question #4 : How To Find An Exponent From A Rational Number

Example Question #5 : How To Find An Exponent From A Rational Number

Example Question #6 : How To Find An Exponent From A Rational Number

Example Question #7 : How To Find An Exponent From A Rational Number

Example Question #8 : How To Find An Exponent From A Rational Number

Example Question #9 : How To Find An Exponent From A Rational Number

Example Question #10 : How To Find An Exponent From A Rational Number

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