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Pre-Algebra : Product Rule of Exponents

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Example Questions

Example Question #1 : Product Rule Of Exponents

Simplify:

$3x^{2}2y^{7}\times 5x^{5}y^{3}$

Possible Answers:

$30x^{10}y^{21}$

$30xy^{17}$

$30x^{7}y^{10}$

$15x^{7}2y^{10}$

$30x^{17}y$

Correct answer:

$30x^{7}y^{10}$

Explanation:

When multiplying variables with exponents, we must remember the Product Rule of Exponents: $x^{m}*x^{n}=x^{m+n}$

Step 1: Reorganize the terms so the terms are together:

$3x^{2}2y^{7}\times 5x^{5}y^{3} = 3\times2\times5\times x^{2} \times x^{5}\times y^{7} \times y^{3}$

Step 2: Multiply $3\times2\times5$ :

$30\times x^{2}\times x^{5}\times y^{7}\times y^{3}$

Step 3: Use the Product Rule of Exponents to combine $x^{2}$ and $x^{5}$ , and then $y^{3}$ and $y^{7}$ :

$30\times x^{7}\times y^{10}$

$30x^{7}y^{10}$

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Example Question #1 : Product Rule Of Exponents

Simplify the following.

$(x^{2})(x^{5})$

Possible Answers:

$x^{10}$

$x^{7}$

$x^{3}$

$x^{-3}$

Correct answer:

$x^{7}$

Explanation:

The Product of Powers Property states when we multiply two powers with the same base, we add the exponents.

In this case, the exponents are 2 and 5

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

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Example Question #2 : Product Rule Of Exponents

Simplify the following expression:

6a^2b(2b^3)

Possible Answers:

12a^2b^3

8a^2 + 8b^4

8a^2b^3

12a^2b^4

Correct answer:

12a^2b^4

Explanation:

The exponent represents how many times the term is being multiplied. So, for example, x^2 means (x)(x) and x^5 would be (x)(x)(x)(x)(x)

So the first term 6a^2b = 6 (a)(a)(b)

And the second term 2b^3 =2(b)(b)(b)

Since the two terms are only separated by parentheses, they are being all multiplied together.

First multiply the coefficients, 6(2)=12

We also have a total of 4 's and 2 's all being multipled together.

The final answer is 12a^2b^4

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Example Question #2 : Product Rule Of Exponents

Simplify the following expression:

(-2t^7s)(-5ts^3)

Possible Answers:

-10t^7s^3

10t^8s^4

-7t^8 + s^4

-7t^7s^3

Correct answer:

10t^8s^4

Explanation:

The exponent represents how many times the term is being multiplied. So, for example, x^2 means (x)(x) and x^5 would be (x)(x)(x)(x)(x)

So the first term -2t^7s = -2(t)(t)(t)(t)(t)(t)(t)(s)

And the second term -5ts^3 = -5(t)(s)(s)(s)

Since the two terms are only separated by parentheses, they are being all multiplied together.

First multiply the coefficients, (-2)(-5)=10

We also have a total of 8 's and 4 's all being multipled together.

The final answer is 10t^8s^4

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Example Question #2 : Product Rule Of Exponents

Simplify the following expression:

(7x^2)(3xy^5)

Possible Answers:

10x^3y^5

x^3 + y^5 + 10

21x^2y^5

21x^3y^5

Correct answer:

21x^3y^5

Explanation:

In the last few problems, we saw one way to multiply terms with exponents.

Another way to explain what we did is to say: "When you MULTIPLY terms together, simplify by ADDING the exponents of each variable."

Here's what that looks like in this case:

First multiply the coefficients: 7(3) = 21

Then ADD the exponents of the variables to simplify. In the first term, the exponent on the is 2. In the second term the exponent is 1. So we ADD 2 + 1 = 3 and have x^3 .

Only the second term has the variable and its exponent is 5. There is nothing to add onto that (because there are no 's in the first term), so it stays y^5 .

Remember, this is all being multiplied together, so the final answer is 21x^3y^5

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Example Question #3 : Product Rule Of Exponents

Simplify the following expression:

(5x^2y^3z^2)(-4xy^6z^2)

Possible Answers:

-20x^2y^1^8z^4

-20x^3y^9z^4

x^2y^1^8z^4

-9x^2y^9z^2

Correct answer:

-20x^3y^9z^4

Explanation:

(5x^2y^3z^2)(-4xy^6z^2)

Remember the rule:

"When you MULTIPLY terms together, simplify by ADDING the exponents of each variable."

Here's what that looks like in this case:

First multiply the coefficients: 5(-4)=-20

Then ADD the exponents of the variables to simplify. In the first term, the exponent on the is 2. In the second term the exponent is 1. So we ADD 2+1=3 and just have x^3 .

In the first term, the exponent on the is 3. In the second term the exponent is 6. So we ADD 3+6=9 and just have y^9 .

In the first term, the exponent on the is 2. In the second term the exponent is 2. So we ADD 2+2=4 and just have z^4 .

Remember, all these parts are being multiplied together, so the final answer is -20x^3y^9z^4

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Example Question #3 : Product Rule Of Exponents

Simplify the following expression:

4y^4(-4xy^2)

Possible Answers:

-16x^8y^8

-16xy^6

-16 + xy^8

Correct answer:

-16xy^6

Explanation:

The exponent represents how many times the term is being multiplied. So, for example, x^2 means (x)(x) and x^5 would be (x)(x)(x)(x)(x)

So the first term 4y^4 = 4(y)(y)(y)(y)

And the second term -4xy^2= -4(x)(y)(y)

Since the two terms are only separated by parentheses, they are being all multiplied together.

First multiply the coefficients, 4(-4) = -16

We also have a total of 6 's and 1 all being multipled together.

The final answer is -16xy^6

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Example Question #7 : Product Rule Of Exponents

Simplify the following expression:

2x^2(-3x^3)

Possible Answers:

-6x - 23

-6x^6

-6x^5

Correct answer:

-6x^5

Explanation:

The exponent represents how many times the term is being multiplied. So, for example, x^2 means (x)(x) and x^5 would be (x)(x)(x)(x)(x)

So the first term 2x^2 = 2(x)(x)

And the second term -3x^3 = -3(x)(x)(x)

Since the two terms are only separated by parentheses, they are being all multiplied together.

First multiply the coefficients, 2(-3) = -6

We also have a total of 's all being multipled together.

The final answer is -6x^5

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Example Question #3 : Product Rule Of Exponents

Simplify:

$\left (5t^{3} \right )^{ 3}$

Possible Answers:

$1 5 t^{9}$

$1 5 t^{6}$

$125 t^{3}$

$125 t^{9}$

$125 t^{6}$

Correct answer:

$125 t^{9}$

Explanation:

$\left (5t^{3} \right )^{ 3} = 5^{ 3} \cdot \left (t^{3} \right ) ^{ 3} = 125 t^{3 \cdot 3} = 125 t^{9}$

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Example Question #4 : Product Rule Of Exponents

Which of the following is equal to $3x^2\times 4x^3$ ?

Possible Answers:

$\small 6x^5$

$\small 7x^5$

$\small 6x^6$

$\small 12x^5$

` $\small 12x^6$

Correct answer:

$\small 12x^5$

Explanation:

Remember that when multiplying variables with exponents, the following property holds true:

$x^a\times x^b = x^{a+b}$

With this knowledge, we can solve the problem:

$3x^2\times 4x^3$

$=(3\times 4)x^2^+^3$

$\small =12x^5$

The answer is $\small 12x^5$ .

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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 #1 : Product Rule Of Exponents

Example Question #1 : Product Rule Of Exponents

Example Question #2 : Product Rule Of Exponents

Example Question #2 : Product Rule Of Exponents

Example Question #2 : Product Rule Of Exponents

Example Question #3 : Product Rule Of Exponents

Example Question #3 : Product Rule Of Exponents

Example Question #7 : Product Rule Of Exponents

Example Question #3 : Product Rule Of Exponents

Example Question #4 : Product Rule Of Exponents

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