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ACT Math : Exponents and the Distributive Property

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Example Questions

Example Question #1 : Exponents And The Distributive Property

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

Possible Answers:

$10x^{2}+4$

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

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

$10x+4$

$25x^{2}+4$

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 #2 : How To Use Foil With Exponents

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

Possible Answers:

64x^3-729

12x-27

12x^3-27

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 : How To Use Foil With Exponents

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 #2 : How To Use Foil With Exponents

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

Possible Answers:

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

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

9x^9-2x^6+7x^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 #4 : How To Use Foil With Exponents

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

Possible Answers:

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

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

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

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

12x^5+4x^4+6x^3+6x^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 #283 : Exponents

Use the FOIL method to simplify the following expression:

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

Possible Answers:

4x^5+4

x^6+2x^4

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

5x^6+4x^4

x^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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Example Question #5 : Exponents And The Distributive Property

The concept of FOIL can be applied to both an exponential expression and to an exponential modifier on an existing expression.

For all , (3x^3 + 7x^2)^2 = __________.

Possible Answers:

9x^3 + 49x^2

9x^6 + 42x^5 + 49x^4

3x^6 + 7x^9

$9x^6 + 21x^{10} + 49x^4$

9x^6 + 49x^4

Correct answer:

9x^6 + 42x^5 + 49x^4

Explanation:

Using FOIL, we see that $(3x^3 + 7x^2)^2 = (3x^3 + 7x^2) \cdot(3x^3 + 7x^2)$

First = $3x^3 \cdot 3x^3 = 9x^6$

Outer = $3x^3 \cdot 7x^2 = 21x^5$

Inner = $7x^2 \cdot 3x^3 = 21x^5$

Last = $7x^2 \cdot 7x^2 = 49x^4$

Remember that terms with like exponents are additive, so we can combine our middle terms:

21x^5 + 21x^5 = 42x^5

Now order the expression from the highest exponent down:

9x^6 + 42x^5 + 49x^4

Thus,

(3x^3 + 7x^2)^2 = 9x^6 + 42x^5 + 49x^4

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Example Question #284 : Exponents

Square the binomial.

$(x^{2}y^{4}+xy^{6})^{2}$

Possible Answers:

$2x^{8}y^{20}$

$x^{4}y^{8}+x^{2}y^{12}$

$x^4y^8+2x^3y^{10}+x^2y^{12}$

$x^4y^{16}+2x^2y^{24}+xy^{36}$

$x^8y^8+x^2y^{12}$

Correct answer:

$x^4y^8+2x^3y^{10}+x^2y^{12}$

Explanation:

$(x^{2}y^{4}+xy^{6})^{2}$

$(x^{2}y^{4}+xy^{6})(x^{2}y^{4}+xy^{6})$

We will need to FOIL.

First: x^2y^4*x^2y^4=x^4y^8

Inside: $xy^6*x^2y^4=x^3y^{10}$

Outside: $x^2y^4*xy^6=x^3y^{10}$

Last: $xy^6*xy^6=x^2y^{12}$

Sum all of the terms and simplify.

$x^4y^8+x^3y^{10}+x^3y^{10}+x^2y^{12}$

$x^4y^8+2x^3y^{10}+x^2y^{12}$

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Example Question #3 : Exponents And The Distributive Property

Simplify:

(xy^2 + 2x^3y^2)(xy^3+3x^2)

Possible Answers:

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

xy^6 + 3x^2y^2 + 2x^3y^6+6x^6y^2

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

11x^1^4y^1^6

3x^4y^5+ x^3y^2 + 2xy^5 + 3x^5y^2

Correct answer:

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

Explanation:

First, merely FOIL out your values. Thus:

(xy^2 + 2x^3y^2)(xy^3+3x^2) becomes

(xy^2* xy^3+ xy^2*3x^2 + 2x^3y^2*xy^3 + 2x^3y^2*3x^2)

Now, just remember that when you multiply similar bases, you add the exponents. Thus, simplify to:

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

Since nothing can be combined, this is the final answer.

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ACT Math : Exponents and the Distributive Property

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Exponents And The Distributive Property

Example Question #2 : How To Use Foil With Exponents

Example Question #1 : How To Use Foil With Exponents

Example Question #2 : How To Use Foil With Exponents

Example Question #4 : How To Use Foil With Exponents

Example Question #283 : Exponents

Example Question #5 : Exponents And The Distributive Property

Example Question #5 : Exponents And The Distributive Property

Example Question #284 : Exponents

Example Question #3 : Exponents And The Distributive Property

All ACT Math Resources

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