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ACT Math : How to find a rational number from an exponent

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Example Questions

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

Which of the following is a value of that satisfies $\log_m256 =4$ ?

Possible Answers:

Correct answer:

Explanation:

When you have a logarithm in the form

$y=\log_bx$ ,

it is equal to

b^y=x .

Using the information given, we can rewrite the given equation in the second form to get

m^4=256 .

Now solving for we get the result.

4^4=256

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Example Question #2 : How To Find A Rational Number From An Exponent

Solve for :

$\log _{a}243 = 5$

Possible Answers:

Correct answer:

Explanation:

When you have a logarithm in the form

$\log _{b}x = y$ ,

it is equal to

b^y = x .

We can rewrite the given equation as

a^5 = 243

Solving for , we get

$\sqrt[5]{a^5} = \sqrt[5]{243}$

a = 3 .

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Example Question #3 : How To Find A Rational Number From An Exponent

Solve for :

$\log_{4}a = 5$

Possible Answers:

1024

10,000

100,000

625

Correct answer:

1024

Explanation:

When you have a logarithm in the form

$\log _{b}x = y$ ,

it is equal to

b^y = x .

We can rewrite the given equation as

4^5 = a

Solving for , we get

a = 1024 .

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

Converting exponents to rational numbers often allows for faster simplification of those numbers.

Which of the following is incorrect? Convert exponents to rational numbers.

Possible Answers:

$18^{\frac{2}{3}} = \sqrt[3]{18^2}$

$16^{\frac{1}{2}} = \sqrt[3]{8}$

$\sqrt[4]{625} = 25^{\frac{1}{2}}$

$\sqrt[5]{10^{10}} = 10^2$

$42^{\frac{0}{3}} = \sqrt[5]{1^{30}}$

Correct answer:

$16^{\frac{1}{2}} = \sqrt[3]{8}$

Explanation:

To identify which answer is incorrect we need to do each of the conversions.

First lets look at $\sqrt[4]{625} = 25^{\frac{1}{2}}$

$625^{\frac{1}{4}}=5$

$25^{\frac{1}{2}}=5$ .

Therefore this conversion is true.

Next lets look at $42^{\frac{0}{3}} = \sqrt[5]{1^{30}}$ . For this particular one we can recognize that anything raised to a zero power is just one therefore this conversion is true.

From here lets look at $16^{\frac{1}{2}} = \sqrt[3]{8}$

$16^{\frac{1}{2}} = \sqrt{16}$

$\sqrt{16} = 4$

$\sqrt[3]{8} = 2$

Thus

$\sqrt{16} \neq \sqrt[3]{8}$ . Therefore this is an incorrect conversion and thus our answer.

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

Sometimes, seeing rational numbers makes it easier to understand an equation.

Convert the following into a rational number or numbers:

$\frac{5^{\frac{1}{3}}\cdot 5^{\frac{4}{5}}}{5^{\frac{1}{15}}}$

Possible Answers:

$\sqrt[4]{5} - 5$

$\sqrt[15]{5^{16}}$

$5(\sqrt[5]{5})$

$\frac{\sqrt[3]{5}}{\sqrt[3]{15}}$

$\sqrt[5]{25}$

Correct answer:

$\sqrt[15]{5^{16}}$

Explanation:

The rule for converting exponents to rational numbers is: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ .

Even with this, it is easier to work the problem as far as we can with exponents, then switch to rational expression when we run out of room:

$\frac{5^{\frac{1}{3}}\cdot 5^{\frac{4}{5}}}{5^{\frac{1}{15}}} = \frac{5^{\frac{17}{15}}}{5^\frac{1}{15}} = 5^{\frac{17}{15}-\frac{1}{15}} = 5^{\frac{16}{15}}$

At last, we convert, and obtain $5^{\frac{16}{15}} = \sqrt[15]{5^{16}}$ .

Thus,

$\frac{5^{\frac{1}{3}}\cdot 5^{\frac{4}{5}}}{5^{\frac{1}{15}}} = \sqrt[15]{5^{16}}$ .

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ACT Math : How to find a rational number from an exponent

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

Example Question #2 : How To Find A Rational Number From An Exponent

Example Question #3 : How To Find A Rational Number From An Exponent

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

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

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