Simplify the expression: [tex]$\left(x^2\right)^5 \cdot\left(x^5\right)^2$[/tex]
Answer (2)
Apply the power of a power rule: ( x 2 ) 5 = x 2 ⋅ 5 = x 10 and ( x 5 ) 2 = x 5 ⋅ 2 = x 10 .
Apply the product of powers rule: x 10 ⋅ x 10 = x 10 + 10 .
Simplify the expression: x 10 + 10 = x 20 .
The simplified expression is x 20 .
Explanation
Understanding the problem We are asked to simplify the expression ( x 2 ) 5 ⋅ ( x 5 ) 2 . To do this, we will use the power of a power rule and the product of powers rule.
Applying the power of a power rule First, we apply the power of a power rule, which states that ( a m ) n = a m ⋅ n . Applying this rule to the first term, we have ( x 2 ) 5 = x 2 ⋅ 5 = x 10 . Applying this rule to the second term, we have ( x 5 ) 2 = x 5 ⋅ 2 = x 10 .
Applying the product of powers rule Now, we substitute these simplified terms back into the original expression: x 10 ⋅ x 10 . Next, we apply the product of powers rule, which states that a m ⋅ a n = a m + n . In our case, we have x 10 ⋅ x 10 = x 10 + 10 = x 20 .
Final Answer Therefore, the simplified expression is x 20 .
Examples
Imagine you are calculating the area of a square where the side length is expressed as a power of x. Simplifying expressions like this helps in determining the area or volume in terms of x, which is useful in various engineering and physics applications, such as calculating the area of solar panels or the volume of a storage tank.
The expression ( x 2 ) 5 ⋅ ( x 5 ) 2 simplifies to x 20 using the power of a power rule and the product of powers rule. First, each term is simplified to x 10 , and then those terms are multiplied to get the final result, x 20 .
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