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ACT Math : Exponential Operations

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Example Questions

Example Question #2 : How To Divide Exponents

Simplify:

$\small \frac{x^2y^3z^4}{x^3y^4z^2}$

Possible Answers:

$\small \frac{yz^2}{x}$

$\small xyz^2$

$\small \frac{z}{x^2y^2}$

$\small \frac{z^2}{xy}$

Correct answer:

$\small \frac{z^2}{xy}$

Explanation:

To simply exponents in a fraction, subtract the exponent for each variable in the denominator from the exponent in the numerator. This will leave you with

$\small x^{-1}y^{-1}z^2$ or $\small \frac{z^2}{xy}$

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Example Question #3 : How To Divide Exponents

What is the value of m where:

n=4

$64^{\frac{n}{12}}=(m)4^{(\frac{1+m}{m+n})}$

Possible Answers:

-2

Correct answer:

Explanation:

If n=4, then 64^(4/12)=64^(1/3)=4. Then, 4=m4^(1+m)/(m+4). If 2 is substituted for m, then 4=24^(1+2)/(2+4)=24^1/2=2√4=22=4.

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Example Question #4 : How To Divide Exponents

Simplify the following:

$\frac{x^7}{x^3}$

Possible Answers:

1/x⁴

x⁴

x^7/3

Cannot be done

Correct answer:

x⁴

Explanation:

These exponents have the same base, x, so they can be divided. To divide them, you take the exponent value in the numerator (the top exponent) and subtract the exponent value of the denominator (the bottom exponent). Here that means we take 7 – 3 so our answer is x⁴.

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Example Question #1 : How To Divide Exponents

Simplify:

$\frac{(2x^{3}y)(4x^{5}y^{4}z)}{x^{2}y}$

Possible Answers:

$8x^{6}y^{4}z$

$8x^{4}y^{5}$

$8x^{6}y^{5}z$

$8x^{13}y^{3}z$

$8x^{4}y^{5}z$

Correct answer:

$8x^{6}y^{4}z$

Explanation:

$a^{m} \cdot a^{n} = a^{m+n}$

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

Use rule for multiplying exponents to simplify the numerator. $\frac{8x^{8}y^{5}z}{x^{2}y}$

Use rule for dividing exponents to simplify.

$\frac{x^{8}}{x^{2}}=x^{8-2}=x^{6}$

$\frac{y^{5}}{y^{1}} = y^{5-1} = y^{4}$

$8x^{6}y^{4}z$

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Example Question #1 : How To Divide Exponents

Simplify:

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2$

Possible Answers:

9a^5b^4

6a^4b^5

36a^6b^8

3a^2b^2

$36a^8b^{10}$

Correct answer:

$36a^8b^{10}$

Explanation:

Simplify:

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2$

Step 1: Simplify the fraction. When dividing exponents subtract the exponents on the bottom from the exponents on the top.

$\left (\frac{18a^3b^3}{3a^{-1}b^{-2}} \right )^2 =\left ( 6a^4b^5 \right )^2$

Step 2: Distribute the exponent. When raising an exponent to a power, multiply them together.

$(6a^4b^5)^2 = 36a^8b^{10}$

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Example Question #6 : How To Divide Exponents

Simplify

$\dpi{100} \small \frac{20x^{4}y^{-3}z^{2}}{5z^{-1}y^{2}x^{2}}=$

Possible Answers:

$\dpi{100} \small 15x^{-2}y^{-2}z^{-2}$

$\dpi{100} \small {4x^{5}y^{-2}}$

$\dpi{100} \small 15x^{2}y^{2}z^{2}$

$\dpi{100} \small \frac{4x^{2}z^{3}}{y^{5}}$

None

Correct answer:

$\dpi{100} \small \frac{4x^{2}z^{3}}{y^{5}}$

Explanation:

Divide the coefficients and subtract the exponents.

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Example Question #1 : Exponential Operations

Which of the following is equal to the expression Equationgre , where

xyz ≠ 0?

Possible Answers:

z/(xy)

1/y

xyz

Correct answer:

1/y

Explanation:

(xy)⁴ can be rewritten as x⁴y⁴ and z⁰ = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.

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Example Question #2 : How To Divide Exponents

If $c\neq 0$ , then $\frac{(c^3)^2}{c^2c^4}=?$

Possible Answers:

c^4

Cannot be determined

c^2

Correct answer:

Explanation:

Start by simplifying the numerator and denominator separately. In the numerator, (c³)² is equal to c⁶. In the denominator, c²* c⁴ equals c⁶ as well. Dividing the numerator by the denominator, c⁶/c⁶, gives an answer of 1, because the numerator and the denominator are the equivalent.

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Example Question #2 : Exponential Operations

If $a\not\equiv0$ , which of the following is equal to $\frac{(a^6*a^2)^3}{a^6}$ ?

Possible Answers:

a¹⁸

a⁴

a⁶

The answer cannot be determined from the above information

Correct answer:

a¹⁸

Explanation:

The numerator is simplified to a^8 (by adding the exponents), then cube the result. a²⁴/a⁶ can then be simplified to $a^{18}$ .

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Example Question #2252 : Act Math

Simplify the following

$\frac{4x^5y^-^4z^4}{2x^-^2y^-^2z^5}$ .

Possible Answers:

$\frac{x^3}{16y^3z}$

$\frac{32x^2y}{z}$

$\frac{2x^3y^3}{z}$

$\frac{y^3z^2}{64x^3}$

$\frac{2x^7}{y^2z}$

Correct answer:

$\frac{2x^7}{y^2z}$

Explanation:

Start by remembering that you "flip" negative exponents over the division bar, thus moving from the top to the bottom and vice-versa. (There are other ways to do this as well, though most students understand this way most easily.)

$\frac{4x^5x^2y^2z^4}{2y^4z^5}$

Next, you just eliminate common factors and combine on the numerator. First combine the s:

$\frac{4x^7y^2z^4}{2y^4z^5}$

Cancel out the numeric coefficient:

$\frac{2x^7y^2z^4}{y^4z^5}$

Now eliminate s and s:

$\frac{2x^7}{y^2z}$

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ACT Math : Exponential Operations

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : How To Divide Exponents

Example Question #3 : How To Divide Exponents

Example Question #4 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #6 : How To Divide Exponents

Example Question #1 : Exponential Operations

Example Question #2 : How To Divide Exponents

Example Question #2 : Exponential Operations

Example Question #2252 : Act Math

All ACT Math Resources

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