Simplify the expression.
[tex]
\begin{array}{l}
(x^2)^3 \\
x^{[?]}
\end{array}
[/tex]
Answer (1)
Apply the power of a power rule: ( x a ) b = x a × b .
Substitute the given values: ( x 2 ) 3 = x 2 × 3 .
Multiply the exponents: 2 × 3 = 6 .
The simplified expression is x 6 , so the missing exponent is 6 .
Explanation
Understanding the Problem We are given the expression ( x 2 ) 3 and asked to simplify it. Our goal is to find the equivalent expression in the form x [ ?] .
Applying the Power of a Power Rule To simplify the expression, we need to use the power of a power rule, which states that when you raise a power to another power, you multiply the exponents: ( x a ) b = x a × b
Substituting the Values In our case, we have a = 2 and b = 3 . Applying the rule, we get: ( x 2 ) 3 = x 2 × 3
Calculating the Exponent Now, we simply multiply the exponents: 2 × 3 = 6
Final Answer Therefore, the simplified expression is: ( x 2 ) 3 = x 6 So, the missing exponent is 6.
Examples
Understanding exponent rules is crucial in many areas, such as calculating compound interest. For example, if you invest money that grows at an annual rate, the formula involves raising the interest rate plus one to the power of the number of years. Simplifying these expressions helps you quickly determine the future value of your investment. Also, in computer science, these rules are used to calculate memory allocation and algorithm complexities.
