Power of a Product Practice Problems - Master Exponent Rules

Practice power of a product problems with step-by-step solutions. Learn to apply exponents to multiplication expressions using (ab)^n = a^n × b^n rule.

📚Master Power of a Product with Interactive Practice
  • Apply the (a×b)^n = a^n×b^n rule to multiplication expressions
  • Solve power of product problems with whole numbers and variables
  • Combine power of product with power of power rules in complex expressions
  • Simplify algebraic expressions with multiple variables and coefficients
  • Work with expressions containing multiple terms and different bases
  • Master coefficient multiplication in algebraic power expressions

Understanding Power of a Product

Complete explanation with examples

When finding an expression with multiplication or an exercise that has only multiplication operations inside a parenthesis and the wholes expression is raised to a certain exponent, we can take the exponent and apply it to each of the terms of the expression or exercise.
We must not forget to keep the multiplication signs between the terms.
Property formula:
(a×b)n=an×bn (a\times b)^n=a^n\times b^n
This property also pertains to algebraic expressions.

Detailed explanation

Practice Power of a Product

Test your knowledge with 39 quizzes

Choose the expression that corresponds to the following:

\( \left(25\times4\right)^3= \)

Examples with solutions for Power of a Product

Step-by-step solutions included
Exercise #1

(2×8×7)2= (2\times8\times7)^2=

Step-by-Step Solution

We begin by using the power rule for parentheses:

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n

That is, the power applied to a product inside parentheses, is applied to each of the terms within, when the parentheses are opened.

We then apply the above rule to the problem:

(287)2=228272 (2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2

Therefore, the correct answer is option d.

Note:

From the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms inside parentheses (as formulated above), then it is also valid for a power over several terms of the product inside parentheses (for example - three terms, etc.).

Answer:

228272 2^2\cdot8^2\cdot7^2

Video Solution
Exercise #2

(2×7×5)3= (2\times7\times5)^3=

Step-by-Step Solution

To solve the problem, we need to apply the power of a product exponent rule. This rule states that when you raise a product to a power, it's the same as raising each factor to that power. In mathematical terms, if you have (abc)n (abc)^n , it is equivalent to an×bn×cn a^n \times b^n \times c^n .

Let's apply this rule step by step:

Our original expression is: (2×7×5)3 (2 \times 7 \times 5)^3 .

We first identify the factors inside the parentheses as 2 2 , 7 7 , and 5 5 .

According to the Power of a Product rule, we can distribute the exponent3 3 to each factor:

First, raise 2 2 to the power of 3 3 to get 23 2^3 .

Then, raise 7 7 to the power of 3 3 to get 73 7^3 .

Finally, raise 5 5 to the power of 3 3 to get 53 5^3 .

Therefore, the expression (2×7×5)3 (2 \times 7 \times 5)^3 simplifies to 23×73×53 2^3 \times 7^3 \times 5^3 .

Answer:

23×73×53 2^3\times7^3\times5^3

Video Solution
Exercise #3

(9×2×5)3= (9\times2\times5)^3=

Step-by-Step Solution

We use the law of exponents for a power that is applied to parentheses in which terms are multiplied:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n

We apply the rule to the problem:

(925)3=932353 (9\cdot2\cdot5)^3=9^3\cdot2^3\cdot5^3

When we apply the power within parentheses to the product of the terms we do so separately and maintain the multiplication,

Therefore, the correct answer is option B.

Answer:

93×23×53 9^3\times2^3\times5^3

Video Solution
Exercise #4

Choose the expression that corresponds to the following:

(2×11)5= \left(2\times11\right)^5=

Step-by-Step Solution

To solve the expression, we can apply the rule for the power of a product, which states that(a×b)n=an×bn \left(a \times b\right)^n = a^n \times b^n .

In this case, our expression is (2×11)5 \left(2\times11\right)^5 , wherea=2 a = 2 and b=11 b = 11 , and n=5 n = 5 .

Applying the power of a product rule gives us:

  • an=25 a^n = 2^5

  • bn=115 b^n = 11^5

Therefore, (2×11)5=25×115 \left(2\times11\right)^5 = 2^5 \times 11^5 .

Answer:

25×115 2^5\times11^5

Video Solution
Exercise #5

Choose the expression that corresponds to the following:

(10×3)4= \left(10\times3\right)^4=

Step-by-Step Solution

To solve this problem, we'll apply the power of a product rule to the expression (10×3)4(10 \times 3)^4.

  • Step 1: Identify the expression.
    The given expression is (10×3)4(10 \times 3)^4.

  • Step 2: Apply the power of a product law.
    According to the rule, (a×b)n=an×bn(a \times b)^n = a^n \times b^n, our expression becomes:
    (10×3)4=104×34(10 \times 3)^4 = 10^4 \times 3^4.

  • Step 3: Evaluate the choices:
    - First choice: 34×1043^4 \times 10^4
    Rearranging terms, this is equivalent to 104×3410^4 \times 3^4. Therefore, it matches our transformed expression.
    - Second choice: 30430^4
    Since 30=10×330 = 10 \times 3, (10×3)4=304(10 \times 3)^4 = 30^4. This simplifies to the same expression.
    - Third choice: 104×3410^4 \times 3^4

    Therefore, the solution is that all answers are correct.

Answer:

All of the above

Video Solution

Frequently Asked Questions

What is the power of a product rule in math?

+
The power of a product rule states that (a×b)^n = a^n×b^n. When you have multiplication inside parentheses raised to an exponent, you can distribute that exponent to each factor while keeping the multiplication sign between terms.

How do you solve (2×3)^4 using power of product rule?

+
Apply the exponent to each factor: (2×3)^4 = 2^4×3^4 = 16×81 = 1,296. This is often easier than calculating (2×3)^4 = 6^4 = 1,296 directly.

Can you use power of product rule with variables and coefficients?

+
Yes! For example, (3x)^2 = 3^2×x^2 = 9x^2. The rule applies to any multiplication expression, whether it contains numbers, variables, or both.

What's the difference between power of product and power of power rules?

+
Power of product: (ab)^n = a^n×b^n (distribute exponent to factors). Power of power: (a^m)^n = a^(mn) (multiply exponents). You often use both rules together in complex expressions.

How do you simplify (2^2×x^3)^2 step by step?

+
First apply power of product: (2^2)^2×(x^3)^2. Then use power of power rule: 2^(2×2)×x^(3×2) = 2^4×x^6 = 16x^6.

When should you simplify inside parentheses first?

+
Simplify first when you have like bases that can be combined. For example, in (2×x×x^2×3)^2, first combine: (6x^3)^2, then apply the power rule: 36x^6.

What are common mistakes with power of product problems?

+
Common errors include: forgetting to apply the exponent to all factors, losing multiplication signs, confusing power of product with power of power rules, and not simplifying coefficients properly.

Does power of product work with more than two factors?

+
Yes! The rule extends to any number of factors: (a×b×c×d)^n = a^n×b^n×c^n×d^n. Apply the outside exponent to each factor individually while maintaining all multiplication operations.

More Power of a Product Questions

Continue Your Math Journey

Practice by Question Type