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ACT Math : Exponents

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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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Example Question #1 : How To Use Scientific Notation

What is the sum of the solutions of the equation x²-2x-8=0?

Possible Answers:

-2

-10

-6

Correct answer:

Explanation:

The equation factors to (x-4)(x+2)=0, indicating that the roots are x=-2,4. The sum of these two roots is 2.

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Example Question #1 : How To Use Scientific Notation

What is the number 350,000,000 in scientific notation?

Possible Answers:

3.5 x 10⁸

3.5 x 10⁶

3.5 x 10⁹

3.5 x 10⁵

3.5 x 10⁷

Correct answer:

3.5 x 10⁸

Explanation:

350,000,000 = 3.5 x 10⁸(exponent is equal to the number of decimal points shifted)

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Example Question #1 : How To Use Scientific Notation

Given the equation 2xy + 4x² - y = 2x, if x = 2, what does y equal?

Possible Answers:

-4

Correct answer:

-4

Explanation:

We plug in the value given for x and solve for y.

2xy + 4x² - y = 2x

2(2)y + 4(2)² - y = 2(2)

4y + 4(4) - y = 4

4y - y + 16 = 4

3y = -12

y = -4

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Example Question #1 : How To Use Scientific Notation

How do you express 657,800 in scientific notation?

Possible Answers:

65.78 × 10⁴

6.578 × 10³

65.78 × 10³

6.578 × 10⁴

65.78 × 10⁶

Correct answer:

65.78 × 10⁴

Explanation:

657800 = 6.578 × 10⁵

657800 = 65.78 × 10⁴

Exponent on 10 is the number of decimal points over the number has shifted.

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Example Question #5 : How To Use Scientific Notation

2 × 10⁰ + 3 × 10² + 4 × 10⁴ = ?

Possible Answers:

40,302

40,320

400,032

4,032

432

Correct answer:

40,302

Explanation:

2 × 10⁰ + 3 × 10² + 4 × 10⁴ =

2 + 300 + 40,000 = 40,302

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Example Question #2362 : Act Math

Give this number in scientific notation:

125,000,000

Possible Answers:

$1.25\times 10^{9}$

$1.25\times 10^{8}$

$1.25\times 10^{7}$

$12.5\times 10^{10}$

$12.5\times 10^{8}$

Correct answer:

$1.25\times 10^{8}$

Explanation:

There are six zeros and two more number places, for a total of eight, so 125,000,000 is equivalent to $1.25 \times 10^{8}$ .

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Example Question #2363 : Act Math

Simplify the following expression:

$(4(i^{2}-3))^{\frac{1}{2}}$

Possible Answers:

Correct answer:

Explanation:

$(4(i^{2}-3))^{\frac{1}{2}} =(4(-1-3))^{\frac{1}{2}} =(4(-4))^{\frac{1}{2}}$

$=(-16))^{\frac{1}{2}}=\sqrt{-16}=\sqrt{(16)(-1)}=4i$

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Example Question #8 : How To Use Scientific Notation

What is the number $\dpi{100} \small 256,000,000,000$ in scientific notation?

Possible Answers:

$\dpi{100} \small 2.56\times 10^{11}$

$\dpi{100} \small 2.56\times 10^{9}$

$\dpi{100} \small 2.56\times 10^{10}$

$\dpi{100} \small 2.56\times 10^{13}$

$\dpi{100} \small 2.56\times 10^{12}$

Correct answer:

$\dpi{100} \small 2.56\times 10^{11}$

Explanation:

$\dpi{100} \small 256,000,000,000 = 2.56\times 10^{11}$

The exponent is equal to the number of decimal points shifted.

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ACT Math : Exponents

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #1 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #5 : How To Use Scientific Notation

Example Question #2362 : Act Math

Example Question #2363 : Act Math

Example Question #8 : How To Use Scientific Notation

All ACT Math Resources

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