ACT Math : Exponential Ratios

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find A Ratio Of Exponents

If \displaystyle x^{2}=25 and \displaystyle y^{4}=81, then which of the following CANNOT be the value of \displaystyle x + y?

 

Possible Answers:

\displaystyle -2

\displaystyle -8

\displaystyle 8

\displaystyle 14

\displaystyle 2

Correct answer:

\displaystyle 14

Explanation:

Even roots of numbers can either be positive or negative. Thus, x = +/- 5 and y = +/- 3. The possible values from x + y can therefore be:

(-5) + (-3) = -8

(-5) + 3 = -2

5 + (-3) = 2

5 + 3 = 8

Example Question #2 : How To Find A Ratio Of Exponents

If \displaystyle \frac{n^{y}}{n^{x}}=n^{6} for all \dpi{100} \small n not equal to 0, which of the following must be true?

Possible Answers:

\dpi{100} \small y-x = 6

\dpi{100} \small yx = 6

\dpi{100} \small \frac{x}{y} = 6

\dpi{100} \small y+x = 6

\dpi{100} \small \frac{y}{x} = 6

Correct answer:

\dpi{100} \small y-x = 6

Explanation:

Remember that \dpi{100} \small \frac{n^{y}}{n^{x}}=n^{y-x}

Since the problem states that \displaystyle \frac{n^{y}}{n^{x}}=n^{6}, you can assume that \dpi{100} \small n^{y-x}=n^{6}

This shows that \dpi{100} \small y-x = 6.

Example Question #3 : How To Find A Ratio Of Exponents

If \displaystyle m and \displaystyle n are positive integers and \displaystyle 4^m=64^n, then what is the value of \displaystyle \frac{m}{n}?

Possible Answers:

\displaystyle \frac{1}{16}

\displaystyle 16

\displaystyle 3

\displaystyle \frac{5}{3}

\displaystyle \frac{1}{3}

Correct answer:

\displaystyle 3

Explanation:

43 = 64

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

Thus, m = 3 and n = 1.

m/n = 3/1 = 3.

Example Question #4 : How To Find A Ratio Of Exponents

Write the following logarithm in expanded form:

 

\displaystyle \log x^{2}y

Possible Answers:

\displaystyle \log x+\log y

\displaystyle 2\log x+\log y

\displaystyle 2\log x-\log y

\displaystyle \log x^{2}+\log y

\displaystyle 2\left ( \log xy \right )

Correct answer:

\displaystyle 2\log x+\log y

Explanation:

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

Example Question #22 : Algebra

If \displaystyle m and \displaystyle n are both rational numbers and \displaystyle 4^{m} = 8^{n}, what is \displaystyle \frac{m}{n}?

Possible Answers:

\displaystyle \frac{4}{1}

\displaystyle \frac{5}{3}

\displaystyle \frac{3}{2}

\displaystyle \frac{2}{1}

Correct answer:

\displaystyle \frac{3}{2}

Explanation:

This question is asking you for the ratio of m to n.  To figure it out, the easiest way is to figure out when 4 to an exponent equals 8 to an exponent.  The easiest way to do that is to list the first few results of 4 to an exponent and 8 to an exponent and check to see if any match up, before resorting to more drastic means of finding a formula.

\displaystyle 4^{1} = 4, 4^{2} = 16, 4^{3} = 64, 4^{4} = 256, 4^{5} = 1024

\displaystyle 8^{1}=8, 8^{2} = 64, 8^{3} = 512, 8^{4} = 4096, 8^{5} = 32768

And, would you look at that. \displaystyle 4^{3}= 8^{2}.  Therefore, \displaystyle \frac{m}{n} = \frac{3}{2}.

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