Use Kelly's steps to simplify this expression.
$\left(5 w^7\right)^2$
What is the simplified power of the product?
$5 w^9$
$10 w^{14}$
$25 w^9$
$25 w^{14}$
Answer (1)
Apply the power of a product rule: ( 5 w 7 ) 2 = 5 2 ( w 7 ) 2 .
Simplify 5 2 = 25 .
Apply the power of a power rule: ( w 7 ) 2 = w 72 = w 14 .
Combine the results: The simplified expression is 25 w 14 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 5 w 7 ) 2 using the power of a product rule. The power of a product rule states that ( ab ) n = a n b n .
Applying the Power of a Product Rule Applying the power of a product rule, we have ( 5 w 7 ) 2 = 5 2 ⋅ ( w 7 ) 2 .
Simplifying the Constant Term Now, we simplify 5 2 . 5 2 = 5 × 5 = 25
Applying the Power of a Power Rule Next, we apply the power of a power rule, which states that ( a m ) n = a m ⋅ n . Therefore, ( w 7 ) 2 = w 7 ⋅ 2 = w 14 .
Combining the Results Combining the results, we have 25 w 14 .
Final Answer Therefore, the simplified expression is 25 w 14 .
Examples
The power of a product rule is useful in various fields, such as physics and engineering, when dealing with scaling relationships. For example, if you are designing a square garden with side length 5 w 7 meters, the area of the garden would be ( 5 w 7 ) 2 = 25 w 14 square meters. Understanding how to simplify such expressions allows you to easily calculate the area for different values of w . This concept extends to more complex scenarios, such as calculating the volume of a cube or the surface area of a sphere, where dimensions are expressed as algebraic terms.
