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

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Example Questions

Example Question #1 : Exponents And Rational Numbers

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 : Exponents And Rational Numbers

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 : Exponents And Rational Numbers

Solve for :

$\log_{4}a = 5$

Possible Answers:

100,000

1024

10,000

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 #171 : Exponents

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:

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

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

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

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

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

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 #172 : Exponents

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:

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

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

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

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

$\sqrt[5]{25}$

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 : Exponents And Rational Numbers

Example Question #2 : Exponents And Rational Numbers

Example Question #3 : Exponents And Rational Numbers

Example Question #171 : Exponents

Example Question #172 : Exponents

All ACT Math Resources

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