Exponents and Rational Numbers

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ACT Math › Exponents and Rational Numbers

Questions 1 - 10

Often, solving a root equation is as simple as switching to exponential form.

Simplify into exponential form:

$\frac{\sqrt[3]{125}}{\sqrt[5]{125}}$

$125^{\frac{2}{15}}$

$25^{\frac{1}{5}}$

$\sqrt[15]{125^{2}}$

$625^{\frac{1}{3}}$

None of these are correct.

Explanation

The rule for exponential ratios is $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ .

Using this, we can convert the numerator and denominator quickly.

$\frac{\sqrt[3]{125}}{\sqrt[5]{125}} = \frac{125^{\frac{1}{3}}}{125^{\frac{1}{5}}}$

Next, we can further simplify by remembering that

$\frac{a^{m}}{a^{n}} = a^{m-n}$ .

Find the least common denominator and simplify:

$\frac{125^{\frac{1}{3}}}{125^{\frac{1}{5}}} = 125^{\frac{5}{15}-\frac{3}{15}} = 125^{\frac{2}{15}}$

Thus, our answer is $125^{\frac{2}{15}}$ . (Remember, the problem asked for exponential form!)

Often, solving a root equation is as simple as switching to exponential form.

Simplify into exponential form:

$\frac{\sqrt[3]{125}}{\sqrt[5]{125}}$

$125^{\frac{2}{15}}$

$25^{\frac{1}{5}}$

$\sqrt[15]{125^{2}}$

$625^{\frac{1}{3}}$

None of these are correct.

Explanation

The rule for exponential ratios is $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ .

Using this, we can convert the numerator and denominator quickly.

$\frac{\sqrt[3]{125}}{\sqrt[5]{125}} = \frac{125^{\frac{1}{3}}}{125^{\frac{1}{5}}}$

Next, we can further simplify by remembering that

$\frac{a^{m}}{a^{n}} = a^{m-n}$ .

Find the least common denominator and simplify:

$\frac{125^{\frac{1}{3}}}{125^{\frac{1}{5}}} = 125^{\frac{5}{15}-\frac{3}{15}} = 125^{\frac{2}{15}}$

Thus, our answer is $125^{\frac{2}{15}}$ . (Remember, the problem asked for exponential form!)

If $\small f(x)=2\times x^{3}$ and $\small g(x)=x+2$ , what is $\small f(g(3))$ ?

Explanation

Start from the inside. $\small g(3)=3+2=5$ . Then, $\small f(5)=2\times 5^{3}=2\times 125=250$ .

If $\small f(x)=2\times x^{3}$ and $\small g(x)=x+2$ , what is $\small f(g(3))$ ?

Explanation

Start from the inside. $\small g(3)=3+2=5$ . Then, $\small f(5)=2\times 5^{3}=2\times 125=250$ .

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.

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.

Find the value of if $\frac{64}{27}=\left ( \frac{4}{3} \right )^{x}$

Explanation

When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent. Therefore, the equation can be rewritten as $\frac{64}{27}=\frac{4^{x}}{3^{x}}$ . From here we can proceed one of two ways. We can either solve x for $64=4^{x}$ or $27=3^{x}$ . Let's solve the first equation. We simply multiply 4 by itself until we reach a value of 64. $4^{1}=4$ , $4^{2}=4\cdot4=16$ , $4^{3}=4\cdot4\cdot4=64$ , and so on. Since $4^{3}=64$ , we know that x = 3.

We can repeat this process for the second equation to get $3^{3}=3\cdot3\cdot3= 27$ , confirming our previous answer. However, since the ACT is a timed test, it is best to only solve one of the equations and move on. Then, if you have time left once all of the questions have been answered, you can come back and double check your answer by solving the other equation.