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

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Example Questions

Example Question #2 : Exponents And Rational Numbers

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

Possible Answers:

$\displaystyle 16$

$\displaystyle 8$

$\displaystyle 4$

$\displaystyle 64$

$\displaystyle 2$

Correct answer:

$\displaystyle 4$

Explanation:

When you have a logarithm in the form

$y=\log_bx$ $\displaystyle y=\log_bx$,

it is equal to

b^y=x $\displaystyle b^y=x$.

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

m^4=256 $\displaystyle m^4=256$.

Now solving for $\displaystyle m$ we get the result.

4^4=256 $\displaystyle 4^4=256$

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

Solve for $\displaystyle a$:

$\log _{a}243 = 5$ $\displaystyle \log _{a}243 = 5$

Possible Answers:

$\displaystyle 4$

$\displaystyle 3$

$\displaystyle 5$

$\displaystyle 9$

$\displaystyle 2$

Correct answer:

$\displaystyle 3$

Explanation:

When you have a logarithm in the form

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

it is equal to

b^y = x $\displaystyle b^y = x$.

We can rewrite the given equation as

$\displaystyle a^5 = 243$

Solving for $\displaystyle a$, we get

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

a = 3 $\displaystyle a = 3$.

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Example Question #2 : Exponents And Rational Numbers

Solve for $\displaystyle a$:

$\log_{4}a = 5$ $\displaystyle \log_{4}a = 5$

Possible Answers:

100,000 $\displaystyle 100,000$

625 $\displaystyle 625$

$\displaystyle 2$

10,000 $\displaystyle 10,000$

1024 $\displaystyle 1024$

Correct answer:

1024 $\displaystyle 1024$

Explanation:

When you have a logarithm in the form

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

it is equal to

b^y = x $\displaystyle b^y = x$.

We can rewrite the given equation as

4^5 = a $\displaystyle 4^5 = a$

Solving for $\displaystyle a$, we get

$\displaystyle a = 1024$.

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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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

Therefore this conversion is true.

Next lets look at $42^{\frac{0}{3}} = \sqrt[5]{1^{30}}$ $\displaystyle 42^{\frac{0}{3}} = \sqrt[5]{1^{30}}$. 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 $16^{\frac{1}{2}} = \sqrt[3]{8}$ $\displaystyle 16^{\frac{1}{2}} = \sqrt[3]{8}$

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

$\sqrt{16} = 4$ $\displaystyle \sqrt{16} = 4$

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

Thus

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

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Example Question #6 : Exponents And Rational Numbers

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

Convert the following into a rational number or numbers:

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

Possible Answers:

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

$\sqrt[5]{25}$ $\displaystyle \sqrt[5]{25}$

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

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

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

Correct answer:

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

Explanation:

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

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:

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

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

Thus,

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

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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 #2 : Exponents And Rational Numbers

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

Example Question #2 : Exponents And Rational Numbers

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

Example Question #6 : Exponents And Rational Numbers

All ACT Math Resources

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