How to find a ratio of exponents - ACT Math

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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

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

Answer

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

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Question

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

Answer

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

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

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

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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

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

Answer

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.

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

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

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

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