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ACT Math : Exponents

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Example Questions

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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Example Question #81 : Exponents

Distribute and simplify: (x^2-1)(x+8)

Possible Answers:

x^3 +7x^2 - 8

10x^2 - x - 8

x^3 +8x^2 - x - 8

x^3 -8x^2 + x + 8

Correct answer:

x^3 +8x^2 - x - 8

Explanation:

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

(x^2-1)(x+8)

Multiply the First terms:

$x^2\cdot x = x^3$

Multiply the Outer terms:

$x^2\cdot 8 = 8x^2$

Multiply the Inner terms:

$-1\cdot x = -x$

Multiply the Last terms:

$-1\cdot 8 = -8$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms). In this case, no two terms are compatible.

Arrange your answer in descending exponential form, and you're done.

(x^2-1)(x+8) = x^3 +8x^2 - x - 8

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Example Question #82 : Exponents

Distribute and simplify: (5x^2-x)(2x+1)

Possible Answers:

10x^3 + 7x^2 - x

10x^3 + 2x^2 + x

10x^3 + 3x^2 - x

10x^2 + 3x + 1

10x^3 - 3x^2 + x

Correct answer:

10x^3 + 3x^2 - x

Explanation:

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

(5x^2-x)(2x+1)

Multiply the First terms:

$5x^2\cdot 2x = 10x^3$

Multiply the Outer terms:

$5x^2\cdot 1 = 5x^2$

Multiply the Inner terms:

$-x\cdot 2x = -2x^2$

Multiply the Last terms:

$-x\cdot 1 = -x$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

5x^2-2x^2 = 3x^2

Arrange your answer in descending exponential form, and you're done.

(5x^2-x)(2x+1) = 10x^3 + 3x^2 - x

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Example Question #83 : Exponents

Distribute and simplify: (x^2+x)(x^2+x)

Possible Answers:

x^4+2x^3+x^2

2x^2+2x+x

2x^4+2x^3+x^2

x^4+2x^3+2x^2

x^4+x^3+x^2+x

Correct answer:

x^4+2x^3+x^2

Explanation:

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

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

Multiply the First terms:

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

Multiply the Outer terms:

$x^2\cdot x = x^3$

Multiply the Inner terms:

$x\cdot x^2 = x^3$

Multiply the Last terms:

$x\cdot x = x^2$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

x^3+x^3 = 2x^3

Arrange your answer in descending exponential form, and you're done.

(x^2+x)(x^2+x) = x^4+2x^3+x^2

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Example Question #84 : Exponents

Distribute and simplify: (4x^5-4x)(4x^5+4x)

Possible Answers:

$16x^{10} + 16x^2$

$16x^{10} +16x^6 - 16x^2$

$8x^{10} + 8x^2$

$8x^{10} - 8x^2$

$16x^{10} - 16x^2$

Correct answer:

$16x^{10} - 16x^2$

Explanation:

A clever eye might spot that this binomial takes the form (a+b)(a-b) , which means you can jump straight to a^2-b^2 as the answer. But let's look at it in detail below.

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

(4x^5-4x)(4x^5+4x)

Multiply the First terms:

$4x^5\cdot 4x^5 = 16x^{10}$

Multiply the Outer terms:

$4x^5\cdot 4x = 16x^6$

Multiply the Inner terms:

$-4x\cdot 4x^5 = -16x^6$

Multiply the Last terms:

$-4x\cdot 4x = -16x^2$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

16x^6-16x^6 = 0

Arrange your answer in descending exponential form, and you're done.

$(4x^5-4x)(4x^5+4x) = 16x^{10} - 16x^2$

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Example Question #85 : Exponents

Distribute and simplify: (2x^4-2x^3)(6x+3x^7)

Possible Answers:

$6x^{11} - 6x^{10} +12x^5 - 12x^4$

$12x^{25} + 6x^{18} - 4x^{10} - 12x^4$

$7x^5 - 4x^{11} + x^{10} - 2x^5$

$6x^{11} - 6x^{10} +12x^5 - 12x^4$

Correct answer:

$6x^{11} - 6x^{10} +12x^5 - 12x^4$

Explanation:

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

(2x^4-2x^3)(6x+3x^7)

Multiply the First terms:

$2x^4\cdot 6x = 12x^5$

Multiply the Outer terms:

$2x^4\cdot 3x^7 = 6x^{11}$

Multiply the Inner terms:

$-2x^3\cdot 6x = -12x^4$

Multiply the Last terms:

$-2x^3\cdot 3x^7 = -6x^{10}$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms). In this case, no two terms are compatible.

Arrange your answer in descending exponential form, and you're done.

$(2x^4-2x^3)(6x+3x^7) = 6x^{11} - 6x^{10} +12x^5 - 12x^4$

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ACT Math : Exponents

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

Example Question #81 : Exponents

Example Question #82 : Exponents

Example Question #83 : Exponents

Example Question #84 : Exponents

Example Question #85 : Exponents

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