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ACT Math : Algebra

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Example Questions

Example Question #711 : Algebra

(b * b⁴* b⁷)^1/2/(b³ * b^x) = b⁵

If b is not negative then x = ?

Possible Answers:

–1

–2

Correct answer:

–2

Explanation:

Simplifying the equation gives b⁶/(b^3+x) = b⁵.

In order to satisfy this case, x must be equal to –2.

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Example Question #712 : Algebra

If〖7/8〗ⁿ= √(〖7/8〗⁵),then what is the value of n?

Possible Answers:

¹/₅

⁵/₂

√5

²/₅

Correct answer:

⁵/₂

Explanation:

7/8 is being raised to the 5th power and to the ¹/₂ power at the same time. We multiply these to find n.

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Example Question #1566 : Gre Quantitative Reasoning

Simplify: (x³ * 2x⁴ * 5y + 4y² + 3y²)/y

Possible Answers:

None of the other answers

10x⁷y + 7y²

10x¹¹ + 7y³

10x⁷ + 7y³

10x⁷ + 7y

Correct answer:

10x⁷ + 7y

Explanation:

Let's do each of these separately:

x³ * 2x⁴ * 5y = 2 * 5 * x³* x⁴* y = 10 * x⁷ * y = 10x⁷y

4y² + 3y² = 7y²

Now, rewrite what we have so far:

(10x⁷y + 7y²)/y

There are several options for reducing this. Remember that when we divide, we can "distribute" the denominator through to each member. That means we can rewrite this as:

(10x⁷y)/y + (7y²)/y

Subtract the y exponents values in each term to get:

10x⁷ + 7y

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Example Question #1562 : Gre Quantitative Reasoning

Compare $5^{27}(850)$ and $5^{29}(34)$ .

Possible Answers:

$\small \small \small 5^{27}(850) = 5^{29}(34)$

The answer cannot be determined from the information given.

$\small \small 5^{27}(850) < 5^{29}(34)$

$\small 5^{27}(850) > 5^{29}(34)$

Correct answer:

$\small \small \small 5^{27}(850) = 5^{29}(34)$

Explanation:

To compare these expressions more easily, we'll change the first expression to have $5^{29}$ in front. We'll do this by factoring out 25 (that is, $\small 5^{2}$ ) from 850, then using the fact that $\small 5^{27}*5^{2}=5^{29}$ .

$5^{27}(850)=5^{27}(25*34)=5^{27}(5^2*34)$

When we combine like terms, we can see that $5^{27}(5^2*34)=5^{29}(34)$ . The two terms are therefore both equal to the same value.

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Example Question #13 : Exponents

Which of the following is equal to $(5^4*5^5)^{20}$ ?

Possible Answers:

$5^{40}$

$5^{29}$

$5^{180}$

$25^{29}$

$25^{180}$

Correct answer:

$5^{180}$

Explanation:

$x^{a} \times x^{b}$ is always equal to $x^{a + b}$ ; therefore, 5 raised to 4 times 5 raised to 5 must equal 5 raised to 9.

$(5^4*5^5)^{20}=(5^{(4+5)})^{20}=(5^9)^{20}$

$(x^{a})^{b}$ is always equal to $x^{(a*b)}$ . Therefore, 5 raised to 9, raised to 20 must equal 5 raised to 180.

$(5^9)^{20}=5^{(9*20)}=5^{180}$

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Example Question #14 : Exponents

Which of the following is equal to ((6^7)(5^7))^7 ?

Possible Answers:

$12^{49}$

$77^{7}$

$30^{49}$

$210^{7}$

$30^{14}$

Correct answer:

$30^{49}$

Explanation:

First, multiply inside the parentheses: (6^7)(5^7)=30^7 .

Then raise to the 7th power: $(30^7)^7=30^{49}$ .

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Example Question #23 : Exponents

Simplify:

$(6x^{2})^{3}\cdot x^{-7}\cdot 2x^{4}$

Possible Answers:

$432x^{3}$

$12x^{2}$

$6x^{2}$

Correct answer:

$432x^{3}$

Explanation:

Remember, we add exponents when their bases are multiplied, and multiply exponents when one is raised to the power of another. Negative exponents flip to the denominator (presuming they originally appear in the numerator).

$(6x^{2})^{3}\cdot x^{-7}\cdot 2x^{4}$

$=\frac{216x^{6}\cdot 2x^{4}}{x^{7}}$

$=\frac{432x^{10}}{x^{7}}$

$=432x^{3}$

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Example Question #24 : Exponents

Evaluate:

$y=\frac{3^{13}-9^{5}}{\left ( \frac{1}{27} \right )^{-3}}$

Possible Answers:

$\dpi{100} \small 78$

$\dpi{100} \small 27$

$\dpi{100} \small 30$

$\dpi{100} \small 81$

$\dpi{100} \small 24$

Correct answer:

$\dpi{100} \small 78$

Explanation:

Can be simplified to:

Capture2

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Example Question #21 : Exponential Operations

Simplify

x^4y^2(3x^4+2yx^2 + 5y) .

Possible Answers:

3x^8y^2 + 7x^4y^3

3x^16+2x^8y^2 + 5x^4y^2

3x^16y^2+2x^8y^2 + 5x^4y^2

3x^8y^2+2x^6y^3 + 5x^4y^3

10x^18y^8

Correct answer:

3x^8y^2+2x^6y^3 + 5x^4y^3

Explanation:

This is just a matter of simply distributing this multiplication. Start by the basic distribution:

x^4y^2(3x^4+2yx^2 + 5y) = 3x^4*x^4y^2+2yx^2*x^4y^2 + 5y*x^4y^2

Now, you just add the exponents that are similar. Thus, you get:

3x^8y^2+2x^6y^3 + 5x^4y^3

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Example Question #21 : How To Multiply Exponents

Simplify the following:

$(a^{x+n})\cdot(a^{x-n})$

Possible Answers:

$a^{x^{2}-n^{2}}$

$(a^{2})^{x^{2}-n^{2}}$

$a^{2x}$

None of these

$\frac{a^{n^{2}}}{a^{x^{2}}}$

Correct answer:

$a^{2x}$

Explanation:

The rule for multiplying exponents is

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

Using this, we see that

$(a^{x+n})\cdot(a^{x-n}) = a^{((x+n)+(x-n))} = a^{2x}$ .

Thus, our answer is $a^{2x}$ .

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #711 : Algebra

Example Question #712 : Algebra

Example Question #1566 : Gre Quantitative Reasoning

Example Question #1562 : Gre Quantitative Reasoning

Example Question #13 : Exponents

Example Question #14 : Exponents

Example Question #23 : Exponents

Example Question #24 : Exponents

Example Question #21 : Exponential Operations

Example Question #21 : How To Multiply Exponents

All ACT Math Resources

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