How to find a rational number from an exponent

Help Questions

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

Questions 1 - 5

1

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

Explanation

When you have a logarithm in the form

$y=\log_bx$ ,

it is equal to

.

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

.

Now solving for we get the result.

2

Solve for :

$\log_{4}a = 5$

Explanation

When you have a logarithm in the form

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

it is equal to

.

We can rewrite the given equation as

Solving for , we get

.

3

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

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

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

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

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

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

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

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

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

4

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

Convert the following into a rational number or numbers:

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

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

$\sqrt[5]{25}$

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

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

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

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,

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

5

Solve for :

$\log _{a}243 = 5$

Explanation

When you have a logarithm in the form

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

it is equal to

.

We can rewrite the given equation as

Solving for , we get

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

.

Return to subject