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PSAT Math : Exponential Ratios and Rational Numbers

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Example Questions

Example Question #11 : Exponential Ratios And Rational Numbers

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 #12 : Exponential Ratios And Rational Numbers

Solve for $\displaystyle x$.

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

Possible Answers:

$\displaystyle 1$

$\displaystyle 3$

$\displaystyle 27$

$\displaystyle 9$

$\displaystyle 81$

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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Example Question #11 : Exponential Ratios And Rational Numbers

$Simplify: \frac{\sqrt[3]{81}}{\sqrt[3]{3}}$ $\displaystyle Simplify: \frac{\sqrt[3]{81}}{\sqrt[3]{3}}$

Possible Answers:

$\sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$ $\displaystyle \sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$

$\frac{\sqrt[3]{81}}{\sqrt[3]{3}} = \frac{3}{1}= 3$ $\displaystyle \frac{\sqrt[3]{81}}{\sqrt[3]{3}} = \frac{3}{1}= 3$

$\frac{\sqrt[3]{9\star 9}}{\sqrt[3]{9}}= \sqrt[3]{9}$ $\displaystyle \frac{\sqrt[3]{9\star 9}}{\sqrt[3]{9}}= \sqrt[3]{9}$

$\frac{81}{3}= 27$ $\displaystyle \frac{81}{3}= 27$

$\sqrt[3]{81\star 3}= \sqrt[3]{243} = 6.24$ $\displaystyle \sqrt[3]{81\star 3}= \sqrt[3]{243} = 6.24$

Correct answer:

$\sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$ $\displaystyle \sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$

Explanation:

$\frac{\sqrt[3]{81}}{\sqrt[3]{3}} = \sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$ $\displaystyle \frac{\sqrt[3]{81}}{\sqrt[3]{3}} = \sqrt[3]{\frac{81}{3}}= \sqrt[3]{27}= 3$

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

Rationalize the denominator:

$\frac{1}{2-\sqrt{3}}$ $\displaystyle \frac{1}{2-\sqrt{3}}$

Possible Answers:

$2+\sqrt{3}$ $\displaystyle 2+\sqrt{3}$

$\sqrt{3}$ $\displaystyle \sqrt{3}$

$\frac{2+\sqrt{3}}{7}$ $\displaystyle \frac{2+\sqrt{3}}{7}$

$\displaystyle 2$

$2-\sqrt{3}$ $\displaystyle 2-\sqrt{3}$

Correct answer:

$2+\sqrt{3}$ $\displaystyle 2+\sqrt{3}$

Explanation:

The conjugate of $2-\sqrt{3}$ $\displaystyle 2-\sqrt{3}$ is $2+\sqrt{3}$ $\displaystyle 2+\sqrt{3}$.

Now multiply both the numerator and the denominator by $2+\sqrt{3}$ $\displaystyle 2+\sqrt{3}$

and you get:

$\frac{2+\sqrt{3}}\left ({2-\sqrt{3}} \right )\left ( 2+\sqrt{3} \right )$ $\displaystyle \frac{2+\sqrt{3}}\left ({2-\sqrt{3}} \right )\left ( 2+\sqrt{3} \right )$

$\left ( 2-\sqrt{3} \right )\left ( 2+\sqrt{3} \right ) =2^{2}-\left ( \sqrt{3} \right )^{2} =4-3 = 1$ $\displaystyle \left ( 2-\sqrt{3} \right )\left ( 2+\sqrt{3} \right ) =2^{2}-\left ( \sqrt{3} \right )^{2} =4-3 = 1$

Hence we get

$\frac{2+\sqrt{3}}{1} = 2+\sqrt{3}$ $\displaystyle \frac{2+\sqrt{3}}{1} = 2+\sqrt{3}$

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

Solve for $\displaystyle x$:

$3^{x-5} = \frac{1}{27}$ $\displaystyle 3^{x-5} = \frac{1}{27}$

Possible Answers:

$\displaystyle 0$

$\displaystyle 3$

$\displaystyle 27$

$\displaystyle 9$

$\displaystyle 2$

Correct answer:

$\displaystyle 2$

Explanation:

$3^{x-5}=\frac{1}{27}=3^{-3}$ $\displaystyle 3^{x-5}=\frac{1}{27}=3^{-3}$

x-5=-3 $\displaystyle x-5=-3$

x=2 $\displaystyle x=2$

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Example Question #12 : Exponential Ratios And Rational Numbers

Solve for $\displaystyle x$.

$\log _{5}x=2$ $\displaystyle \log _{5}x=2$

Possible Answers:

$\displaystyle 2$

$\displaystyle 25$

125 $\displaystyle 125$

$\displaystyle 5$

$\displaystyle 10$

Correct answer:

$\displaystyle 25$

Explanation:

$log_5{x} = 2$ $\displaystyle log_5{x} = 2$

$x = 5^{2} = 25$ $\displaystyle x = 5^{2} = 25$

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

If,

$log_a{120}= log_b{120}$ $\displaystyle log_a{120}= log_b{120}$

What does a=? $\displaystyle a=?$

Possible Answers:

b=10 $\displaystyle b=10$

ab=10 $\displaystyle ab=10$

a=b $\displaystyle a=b$

$\frac{a}{b}= 120$ $\displaystyle \frac{a}{b}= 120$

a=10 $\displaystyle a=10$

Correct answer:

a=b $\displaystyle a=b$

Explanation:

If $log_a{x} = log_b{x}$ $\displaystyle log_a{x} = log_b{x}$,

then a=b $\displaystyle a=b$.

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

$\sqrt{x+3} -x = 1$ $\displaystyle \sqrt{x+3} -x = 1$

Possible Answers:

x=0 $\displaystyle x=0$

x=9 $\displaystyle x=9$

x=4 $\displaystyle x=4$

x=13 $\displaystyle x=13$

x=1 $\displaystyle x=1$

Correct answer:

x=1 $\displaystyle x=1$

Explanation:

From the equation in the problem statement

$\sqrt{x+3} = x + 1$ $\displaystyle \sqrt{x+3} = x + 1$

Now squaring both sides we get

$x+3=x^{2}+2x+1$ $\displaystyle x+3=x^{2}+2x+1$ this is a quadratic equation which equals

$x^{2}+x -2 = 0$ $\displaystyle x^{2}+x -2 = 0$

and the factors of this equation are

$\left ( x+2 \right )\left ( x-1 \right )$ $\displaystyle \left ( x+2 \right )\left ( x-1 \right )$

This gives us $x=-2\ or\ x=1$ $\displaystyle x=-2\ or\ x=1$.

However, if we plug these solutions back into the original equation, $\displaystyle -2$ does not create an equality. Therefore, x=-2 $\displaystyle x=-2$ is an extraneous solution.

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

For some positive integer $\displaystyle x$, if x^3=27 $\displaystyle x^3=27$, what is the value of (x+2)^3 $\displaystyle (x+2)^3$?

Possible Answers:

$\displaystyle 64$

125 $\displaystyle 125$

$\displaystyle -25$

$\displaystyle 25$

$\displaystyle 8$

Correct answer:

125 $\displaystyle 125$

Explanation:

If x^3=27 $\displaystyle x^3=27$, then $\displaystyle x$ must equal 3 (Note that $\displaystyle x$ cannot be -3 because you need it to be positive.

Now, plug x=3 $\displaystyle x=3$ into the new equation (x+2)^3 $\displaystyle (x+2)^3$:

$\displaystyle =(3+2)^3$

=5^3 $\displaystyle =5^3$

=125 $\displaystyle =125$

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PSAT Math : Exponential Ratios and Rational Numbers

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #11 : Exponential Ratios And Rational Numbers

Example Question #12 : Exponential Ratios And Rational Numbers

Example Question #11 : Exponential Ratios And Rational Numbers

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

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

Example Question #12 : Exponential Ratios And Rational Numbers

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

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

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

All PSAT Math Resources

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