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ACT Math : Algebra

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Example Questions

Example Question #11 : How To Divide Exponents

What is the simplified form for the following expression?

$\frac{(2xz^2)^3}{(4zx^2)^2}$

Possible Answers:

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

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

2*z^3*x^3

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

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

Correct answer:

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

Explanation:

Break up by variable.

$\frac{2^3}{4^2}= \frac{1}{2}$

$\frac{x^3}{x^4}= \frac{1}{x}$

$\frac{z^6}{z^2} = z^4$

Therefore the simplified form becomes,

$\frac{(2xz^2)^3}{(4zx^2)^2}=\frac{z^4}{2x}$ .

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Example Question #771 : Algebra

$64^{\frac{4}{3}}$ can be written as which of the following?

A. $\sqrt[3]{64^4}$

B. $(\sqrt[3]{64})^{4}$

C. 256

Possible Answers:

B and C

C only

A, B and C

A and C

B only

Correct answer:

A, B and C

Explanation:

C is computing the exponent, while A and B are equivalent due to properties of fractional exponents.

Remember that...

$a^{\frac{b}{c}} = \sqrt[c]{a^b} = (\sqrt[c]{a})^b$

$64^{\frac{4}{3}}=$ $\sqrt[3]{64^4}=$ $(\sqrt[3]{64})^{4}=$ 256

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Example Question #568 : Algebra

$\dpi{100} \small \frac{7^{10}-7^{8}}{7^{9}-7^{7}}=$

Possible Answers:

$\dpi{100} \small 343$

$\dpi{100} \small 49$

$\dpi{100} \small 28$

$\dpi{100} \small 42$

$\dpi{100} \small 7$

Correct answer:

$\dpi{100} \small 7$

Explanation:

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is 7^7 .

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}$

Now, we can cancel out the 7^7 from the numerator and denominator and continue simplifying the expression.

$\frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}=\frac{7^3-7^1}{7^2-1}=\frac{343-7}{49-1}=\frac{336}{48}=7$

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Example Question #1 : How To Use Foil With Exponents

For all $x,\ (5x+2)^{2}=$ ?

Possible Answers:

$10x^{2}+4$

$25x^{2}+20x+4$

$25x^{2}+10x+4$

$25x^{2}+4$

$10x+4$

Correct answer:

$25x^{2}+20x+4$

Explanation:

$(5x+2)^{2}$ is equivalent to $(5x+2)(5x+2)$ .

Using the FOIL method, you multiply the first number of each set $5x\cdot 5x=25x^{2}$ , multiply the outer numbers of each set $5x\cdot 2=10x$ , multiply the inner numbers of each set $2\cdot 5x=10x$ , and multiply outer numbers of each set $2\cdot 2=4$ .

Adding all these numbers together, you get $25x^{2}+10x+10x+4=25x^{2}+20x+4$ .

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Example Question #72 : Exponents

$(4x-9)^{3}=?$

Possible Answers:

12x^3-27

64x^3-729

12x-27

64x^3-432x^2+972x-729

Correct answer:

64x^3-432x^2+972x-729

Explanation:

FOIL the first two terms:

(4x-9)(4x-9)=16x^2-36x-36x+81 = 16x^2-72x+81

Next, multiply this expression by the last term:

(16x^2-72x+81)(4x-9)=64x^3-144x^2-22x^2+648x+324x-729

Finally, combine the terms:

(4x-9)^3=64x^3-432x^2+972x-729

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Example Question #1 : Exponents And The Distributive Property

If q=2 , what is the value of the equation $q(q-7)^{2}$ ?

Possible Answers:

100

Correct answer:

Explanation:

Plug in for in the equation $q(q-7)^{2}$

That gives: $2(2-7)^{2}$

Then solve the computation inside the parenthesis: $2(-5)^{2}$

The answer should then be

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Example Question #4 : Exponents And The Distributive Property

The expression (4x^3-2)(9x^6+7) is equivalent to __________.

Possible Answers:

36x^9-18x^6+28x^3-14

9x^9-2x^6+7x^3-14

4x^9-18x^6+28x^3-14

$36x^{18}-18x^6+28x^3-14$

36x^9-18x^6-28x^3+14

Correct answer:

36x^9-18x^6+28x^3-14

Explanation:

Use FOIL and be mindful of exponent rules. Remember that when you multiply two terms with the same bases but different exponents, you will need to add the exponents together.

Foilexpo

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Example Question #5 : Exponents And The Distributive Property

The expression (3x^2+x)(4x^3+2x) is equivalent to __________.

Possible Answers:

12x^5+4x^4+8x^2

12x^6+4x^3+6x^2+2x

12x^5+4x^4+6x^3+2x^2

12x^5+4x^4+6x^3+6x^2

12x^5+10x^4+6x^3+2x^2

Correct answer:

12x^5+4x^4+6x^3+2x^2

Explanation:

Remember to add exponents when two terms with like bases are being multiplied.

Foilexpo

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Example Question #3 : How To Use Foil With Exponents

Use the FOIL method to simplify the following expression:

(x^3+2x^2)^2

Possible Answers:

x^6+2x^4

4x^5+4

x^6+4x^5+4x^4

x^6+4x^4

5x^6+4x^4

Correct answer:

x^6+4x^5+4x^4

Explanation:

Use the FOIL method to simplify the following expression:

(x^3+2x^2)^2

Step 1: Expand the expression.

(x^3+2x^2)(x^3+2x^2)

Step 2: FOIL

First: $x^3\cdot x^3 = x^6$

Outside: $x^3 \cdot 2x^2 =2x^5$

Inside: $2x^2 \cdot x^3 = 2x^5$

Last: $2x^2 \cdot 2x^2 = 4x^4$

Step 2: Sum the products.

x^6+2x^5+2x^5+4x^4

x^6+4x^5+4x^4

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Example Question #5 : Exponents And The Distributive Property

The rule for adding exponents is $a^m + a^n = a^{m+n}$ .

The rule for multiplying exponents is $(a^m)^n = a^{m\cdot n}$ .

Terms with matching variables AND exponents are additive.

Multiply: $(a^3 + b^2c^2) \cdot (a^7 + bc^3)$

Possible Answers:

$a^{21}+b^2c^6$

$a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5$

$a^{10} + b^3c^5$

$a^{10} + a^{10}b^3c^5 + b^3c^5$

$a^{10} + a^21b^2c^6 + b^3c^5$

Correct answer:

$a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5$

Explanation:

Using FOIL on $(a^3 + b^2c^2) \cdot (a^7 + bc^3)$ , we see that:

First: $a^3 \cdot a^7 = a^{10}$

Outer: $a^3 \cdot bc^3 = a^3bc^3$

Inner: $b^2c^2 \cdot a^7 = a^7b^2c^2$

Last: $b^2c^2 \cdot bc^3 = b^3c^5$

Note that the middle terms are not additive: while they share common variables, they do not share matching exponents.

Thus, we have $a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5$ . The arrangement goes by highest leading exponent, and alphabetically in the case of the last two terms.

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #11 : How To Divide Exponents

Example Question #771 : Algebra

Example Question #568 : Algebra

Example Question #1 : How To Use Foil With Exponents

Example Question #72 : Exponents

Example Question #1 : Exponents And The Distributive Property

Example Question #4 : Exponents And The Distributive Property

Example Question #5 : Exponents And The Distributive Property

Example Question #3 : How To Use Foil With Exponents

Example Question #5 : Exponents And The Distributive Property

All ACT Math Resources

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