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ACT Math : Algebra

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Example Questions

Example Question #1 : How To Subtract Exponents

Reduce $\frac {x^9y^5}{x^3y^2z}$ to simplest form.

Possible Answers:

x^6y^3z

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

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

$\frac {x^6y^3}{z}$

x^3y^3z

Correct answer:

$\frac {x^6y^3}{z}$

Explanation:

When dividing terms with the same bases but different exponents, you will need to subtract all the pertinent exponents.

$\frac {x^9}{x^3}$ becomes x^6 ,

$\frac {y^5}{y^2}$ becomes y^3 ,

and $\frac {1}{z}$ stays the same because there is no other z term to combine with it.

Thus resulting in:

$\frac{x^6y^3}{z}$

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Example Question #1 : How To Subtract Exponents

Simplify: 3² * (4²³ - 4²¹)

Possible Answers:

3^3 * 4^21

None of the other answers

3^3 * 4^21 * 5

3^21

4^4

Correct answer:

3^3 * 4^21 * 5

Explanation:

Begin by noting that the group (4²³ - 4²¹) has a common factor, namely 4²¹. You can treat this like any other constant or variable and factor it out. That would give you: 4²¹(4² - 1). Therefore, we know that:

3² * (4²³ - 4²¹) = 3² * 4²¹(4² - 1)

Now, 4² - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:

3² * 4²¹(4² - 1) = 3² * 4²¹(3 * 5)

Combining multiples withe same base, you get:

3³ * 4²¹ * 5

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Example Question #2 : How To Subtract Exponents

Simplify. Leave no negative exponents in the final answer.

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3}$

Possible Answers:

$\frac{y^3}{z^5}$

$a^{6}y^{5}z^{5}$

None of these are the correct answer.

$\frac{y^5}{z^{5}}$

$-\frac{y^{5}}{z^{5}}$

Correct answer:

$\frac{y^5}{z^{5}}$

Explanation:

The first step in the problem is to combine like terms in the numerator, remembering that $a^{m}\cdot a^{n} = a^{m+n}$ :

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3}$

Next, we resolve the numerator, using $\frac{a^{m}}{a^{n}}=a^{m-n}$ and $a^{0} =1$ :

$\frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3} = y^{5}z^{-5}$

Lastly, simplify the negative exponent using $a^{-m}=\frac{1}{a^{m}}$

$y^{5}z^{-5} = \frac{y^{5}}{z^{5}}$

Thus,

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{y^{5}}{z^{5}}$

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Example Question #1 : How To Subtract Exponents

Simplify to remove fractions:

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}}$

Possible Answers:

$w^{-2}x^{9}y^{2}z$

None of these are correct.

$w^{-3}y^{2}z$

$w^{3}x^{3}y^{2}z^3$

$w^{2}y^{-4}z^{-1}$

Correct answer:

$w^{-3}y^{2}z$

Explanation:

The first step is to simplify each fraction by dividing like terms, remembering that $\frac{a^{m}}{a^{n}}=a^{m-n}$ , to get:

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = (w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5})$

Next, combine using multiplication and the rule $a^{m}\cdot a^{n}= a^{m+n}$ :

$(w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5}) = w^{-3}y^{2}z$

Since the problem specifies that we must avoid fractions, we will not eliminate the negative exponents.

So,

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = w^{-3}y^{2}z$

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Example Question #2 : How To Subtract Exponents

Simplify the following: $\frac{x^{10}}{x^{6}}$

Possible Answers:

$x^{-4}$

$x^{16}$

$x^{4}$

$x^{\frac{10}{6}}$

Correct answer:

$x^{4}$

Explanation:

When dividing exponential expressions with the same base, subtract the exponents. In this problem, the exponents are and . When subtracted, the result is . Thus, the correct answer is $x^{4}$ .

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Example Question #1 : How To Subtract Exponents

$\frac{10^{6}}{10^{2}}$ can be written as which of the following?

A. $10^{\frac{6}{2}}$

B. $10^{4}$

C. 10000

Possible Answers:

A, B and C

B only

B and C

C only

A and C

Correct answer:

B and C

Explanation:

A is not equivalent because exponents in denominators mean subtraction of exponents and not division of them. Furthermore, A, when computed, comes out to 1000 instead of 10000 .

B is equivalent by the aforementioned exponential property, while C is simply computing the expression.

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Example Question #1 : How To Subtract Exponents

Simplify: $\frac{x^{3}}{x}$

Possible Answers:

$x^{3}$

$x^{2}$

Correct answer:

$x^{2}$

Explanation:

When two exponents with the same base are being divided, subtract the exponent of the denominator from the exponent of the numerator to yield a new exponent. Attach that exponent to the base, and that is your answer.

In this case, subtract from . That yields as the new exponent and $x^{2}$ as the answer.

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Example Question #1 : How To Divide Exponents

Simplify

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

When working with polynomials, dividing is the same as multiplying by the reciprocal. After multiplying, simplify. The correct answer for division is

and the correct answer for multiplication is

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Example Question #1 : How To Divide Exponents

Simplify:

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

Possible Answers:

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

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

$\small xyz^2$

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

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 #61 : 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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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #2 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #2 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #61 : Exponents

All ACT Math Resources

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