Exponential Ratios and Rational Numbers - PSAT Math

Card 0 of 19

Question

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

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:

$\log x^{2}y$

Answer

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

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Question

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

Answer

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

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Question

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

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:

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

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 :

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

Answer

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

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Question

Solve for .

$\log _{5}x=2$

Answer

$log_5{x} = 2$

$x = 5^{2} = 25$

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Question

If,

$log_a{120}= log_b{120}$

What does

Answer

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

then .

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Question

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

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

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?

Answer

The answer is 33 = 27

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Question

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

Answer

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Question

Solve for .

$2^{x}= 64$

Answer

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

Hence

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Question

Simplify:

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

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 :

$\sqrt{x} -1= 4$

Answer

From the equation one can see that

$\sqrt{x} = 5$

Hence must be equal to 25.

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Question

Evaluate:

$\log 0.1=$

Answer

$0.1= 10^{-1}$

$\log 0.1=-1$

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Question

Solve for .

$\log_{2}x=4$

Answer

$x= 2^4{}= 16$

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Question

Solve for .

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

Answer

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

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

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Question

Solve for .

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

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 $\log 15$

Answer

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

Hence the correct answer is .

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