Simplify the expression.
[tex]\begin{array}{l}
\left(x^4\right)^3 \\
x^{[?]}
\end{array}[/tex]
Answer (1)
Use the power of a power rule: ( x a ) b = x a ⋅ b .
Apply the rule to the given expression: ( x 4 ) 3 = x 4 ⋅ 3 .
Calculate the exponent: 4 ⋅ 3 = 12 .
The simplified expression is x 12 .
Explanation
Understanding the Problem We are given the expression ( x 4 ) 3 and asked to simplify it. Our goal is to find the exponent of x after simplification.
Applying the Power of a Power Rule To simplify the expression, we will use the power of a power rule, which states that when you raise a power to a power, you multiply the exponents: ( x a ) b = x a ⋅ b
Calculating the Exponent Applying this rule to our expression, we have: ( x 4 ) 3 = x 4 ⋅ 3
Final Answer Now, we multiply the exponents: 4 ⋅ 3 = 12 Therefore, the simplified expression is x 12 .
Examples
Understanding exponent rules is crucial in many fields, such as computer science when dealing with data storage sizes (e.g., kilobytes, megabytes, gigabytes) or in physics when calculating quantities that scale exponentially, like radioactive decay. For instance, if a data storage doubles every year, the size after several years can be calculated using exponential growth, similar to the power rule used here.
