Exponential Ratios and Rational Numbers - PSAT Math

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Question

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

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Answer

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Question

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?

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Answer

The answer is 33 = 27

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Question

If and are positive integers and , then what is the value of $\frac{m}{n}$ ?

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Answer

43 = 64

Alternatively written, this is 4(4)(4) = 64 or 43 = 641.

Thus, m = 3 and n = 1.

m/n = 3/1 = 3.

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Question

Write the following logarithm in expanded form:

$\log x^{2}y$

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Answer

$\log x^{2}y=\log x^{2}+\log y=2\log x+\log y$

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Question

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

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Answer

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

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Question

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

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Answer

From the equation in the problem statement

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

Now squaring both sides we get

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

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

and the factors of this equation are

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

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

However, if we plug these solutions back into the original equation, does not create an equality. Therefore, is an extraneous solution.

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Question

Rationalize the denominator:

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

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Answer

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

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

and you get:

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

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

Hence we get

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

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Question

Solve for :

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

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Answer

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

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Question

Solve for .

$\log _{5}x=2$

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Answer

$log_5{x} = 2$

$x = 5^{2} = 25$

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Question

If,

$log_a{120}= log_b{120}$

What does

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Answer

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

then .

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Question

For some positive integer , if , what is the value of ?

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Answer

If , then must equal 3 (Note that cannot be -3 because you need it to be positive.

Now, plug into the new equation :

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Question

Solve for .

$2^{x}= 64$

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Answer

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

Hence

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Question

Simplify:

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

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Answer

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

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Question

Solve for :

$\sqrt{x} -1= 4$

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Answer

From the equation one can see that

$\sqrt{x} = 5$

Hence must be equal to 25.

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Question

Evaluate:

$\log 0.1=$

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Answer

$0.1= 10^{-1}$

$\log 0.1=-1$

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Question

Solve for .

$\log_{2}x=4$

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Answer

$x= 2^4{}= 16$

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Question

Solve for .

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

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Answer

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

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

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Question

Solve for .

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

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Answer

Use the rules of logarithms to combine terms.

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

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

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

By fatoring we get

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

However, you cannot take the logarithm of a negative number. Thus, the only value for is .

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Question

$\log 3=a$

and

$\log 5=b$

Find $\log 15$

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Answer

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

Hence the correct answer is .

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Question

Write the following logarithm in expanded form:

$\log x^{2}y$

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Answer

$\log x^{2}y=\log x^{2}+\log y=2\log x+\log y$

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