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Pre-Algebra : Product Rule of Exponents

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Example Questions

Example Question #31 : Product Rule Of Exponents

Simplify: $4^2 \cdot 4^{-5} \cdot 4^{10}$

Possible Answers:

$4^{-3}$

4^4

4^7

$4^{-4}$

$4^{-7}$

Correct answer:

4^7

Explanation:

Since the bases are all alike and are multiplied, the exponents can be added.

In mathematical terms:

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

Plugging in four as our base and the various exponents we get the following,

$4^2 \cdot 4^{-5} \cdot 4^{10} = 4^{2-5+10} = 4^7$ .

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Example Question #32 : Product Rule Of Exponents

Expand: (3x^3)^3

Possible Answers:

27 x^6

27 x^9

9x^9

3x^9

3x^6

Correct answer:

27 x^9

Explanation:

The exponents can be multiplied in this case.

Expand out the terms to verify.

(3x^3)^3 = (3x^3)(3x^3)(3x^3) = 27 x^9

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Example Question #123 : Polynomials

Solve: 3(a^2)^3 + (3a^2)^3

Possible Answers:

27a^6

6a^6

30a^6

9a^6

12a^6

Correct answer:

30a^6

Explanation:

Evaluate the quantities inside the parentheses first. The powers are multiplied when they are bounded in the parentheses.

3(a^2)^3 + (3a^2)^3 = 3a^6 + 3^3(a^6)

3a^6 + 3^3(a^6) = 3a^6 + 27a^6

Combine like-terms.

3a^6 + 27a^6 = 30a^6

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Example Question #130 : Polynomials

Evaluate:

$(a^2)^4\cdot(a^{-2})^3$

Possible Answers:

$a^{24}$

$\frac{1}{a^2}$

a^2

Correct answer:

a^2

Explanation:

Simplify each parentheses before multiplying.

Recall the rules of exponents which states if exponents are being raised to another one they are multiplied and if like bases are being multiplied their exponents are added.

$(a^n)^m=a^{n\cdot m}$

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

Applying these rules we get,

$(a^2)^4\cdot(a^{-2})^3 = a^8 \cdot a^{-6} = a^{8+(-6)}=a^2$ .

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Example Question #33 : Product Rule Of Exponents

Simplify: -b(b^2)^3

Possible Answers:

b^5-b

-b^7

b^7-b

-b^6

b^6-b

Correct answer:

-b^7

Explanation:

Evaluate the powers by multiplication.

-b(b^2)^3 = -b (b^6)

Use the power rule to combine the terms. The exponents can then be added.

-b (b^6) = -b^7

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Example Question #35 : Product Rule Of Exponents

Solve: $(x^3)^2 \cdot (x^2)^3$

Possible Answers:

$x^{12}$

2x^5

$x^{10}$

$x^{25}$

2x^6

Correct answer:

$x^{12}$

Explanation:

Simplify the first two terms using the product rule of exponents. When the base of some power is squared, we can simply multiply the two powers together.

$(x^3)^2 \cdot (x^2)^3 = x^{3\times 2} \cdot x^{2\times 3}$

Simplify the powers.

$x^{3\times 2} \cdot x^{2\times 3} =x^6 \cdot x^6$

Use the power rule to add the powers since the bases are alike.

$x^6 \cdot x^6 = x^{6+6} = x^{12}$

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Example Question #36 : Product Rule Of Exponents

Solve

(5x^2)(-3x^4)

Possible Answers:

-15x^6

2x^8

15x^6

2x^6

-15x^8

Correct answer:

-15x^6

Explanation:

If a variable has an exponent with the value of a, it means we are taking that variable and mulitplying it by itself a times. For example,

x^2 = x*x

x^3 = x*x*x

and so on.

So when we multiply

(5x^2)(-3x^4)

we can expand it to look like

(5*x*x)(-3*x*x*x*x)

If we multiply, we get

-15*x*x*x*x*x*x

which equals

-15x^6

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Example Question #34 : Product Rule Of Exponents

Simplify the following:

$x^5 \cdot x^2$

Possible Answers:

x^7

$x^{52}$

x^3

x^2

$x^{10}$

Correct answer:

x^7

Explanation:

When you multiply two terms together with the same base, you add the exponents. So,

$x^5 \cdot x^2 = x^{5 + 2} = x^7$

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Example Question #38 : Product Rule Of Exponents

Simplify the following

$x^4 \cdot y^2 \cdot x^{-2} \cdot y$

Possible Answers:

$x^2 \cdot y^3$

$x^2 \cdot y^2$

$2x^2 \cdot 2y^3$

$x^{-8} \cdot y^2$

$x^6 \cdot y^2$

Correct answer:

$x^2 \cdot y^3$

Explanation:

When multiplying terms with the same base, we add the exponents. Therefore,

$x^4 \cdot y^2 \cdot x^{-2} \cdot y$

can be written as

$x^{4+ -2} \cdot y^{2 + 1}$

$x^2 \cdot y^3$

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Example Question #35 : Product Rule Of Exponents

Simplify the following expression:

6x^3*8x^7

Possible Answers:

$48x^{21}$

$14x^{21}$

$14x^{10}$

$48x^{10}$

Correct answer:

$48x^{10}$

Explanation:

Simplify the following expression:

6x^3*8x^7

So, we are multiplying exponents. When doing so, we need to treat the numbers out in front like regular multiplication, and add the exponents:

$(6*8)x^{3+7}=48x^{10}$

So we have our answer:

$48x^{10}$

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Pre-Algebra : Product Rule of Exponents

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #31 : Product Rule Of Exponents

Example Question #32 : Product Rule Of Exponents

Example Question #123 : Polynomials

Example Question #130 : Polynomials

Example Question #33 : Product Rule Of Exponents

Example Question #35 : Product Rule Of Exponents

Example Question #36 : Product Rule Of Exponents

Example Question #34 : Product Rule Of Exponents

Example Question #38 : Product Rule Of Exponents

Example Question #35 : Product Rule Of Exponents

All Pre-Algebra Resources

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