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ACT Math : Exponents

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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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Example Question #1 : How To Use Scientific Notation

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

Possible Answers:

-10

-2

-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 10⁹

3.5 x 10⁶

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 #3 : 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

65.78 × 10³

65.78 × 10⁶

6.578 × 10⁴

6.578 × 10³

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

4,032

400,032

432

Correct answer:

40,302

Explanation:

2 × 10⁰ + 3 × 10² + 4 × 10⁴ =

2 + 300 + 40,000 = 40,302

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Example Question #1 : How To Use Scientific Notation

Give this number in scientific notation:

125,000,000

Possible Answers:

$1.25\times 10^{7}$

$12.5\times 10^{10}$

$1.25\times 10^{8}$

$1.25\times 10^{9}$

$12.5\times 10^{8}$

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 #171 : Exponents

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 #2 : Other Exponents

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

Possible Answers:

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

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

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

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

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

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 #171 : Exponents

Example Question #172 : Exponents

Example Question #1 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #3 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #5 : How To Use Scientific Notation

Example Question #1 : How To Use Scientific Notation

Example Question #171 : Exponents

Example Question #2 : Other Exponents

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