Write the expression as an exponent with the base of a: [tex]$\left(a^4\right)^3$[/tex]
Answer (1)
Apply the power of a power rule: ( a m ) n = a m \tcdot n .
Multiply the exponents: 4 \tcdot 3 = 12 .
Simplify the expression: ( a 4 ) 3 = a 12 .
The expression as an exponent with the base of a is a 12 .
Explanation
Understanding the Problem We are asked to write the expression ( a 4 ) 3 as an exponent with the base of a . This involves using the power of a power rule in exponents.
Stating the Power of a Power Rule The power of a power rule states that when you raise a power to another power, you multiply the exponents. Mathematically, this is expressed as ( a m ) n = a m ⋅ n , where a is the base and m and n are the exponents.
Applying the Rule to the Expression Applying this rule to the given expression ( a 4 ) 3 , we have a as the base, 4 as the inner exponent, and 3 as the outer exponent. Therefore, we multiply the exponents 4 and 3 .
Simplifying the Exponent Multiplying the exponents, we get 4 ⋅ 3 = 12 . Thus, the expression ( a 4 ) 3 simplifies to a 12 .
Final Answer Therefore, the expression ( a 4 ) 3 written as an exponent with the base of a is a 12 .
Examples
Understanding exponent rules is crucial in many scientific and engineering applications. For instance, when calculating the volume of a cube with side length a 4 , and then cubing that volume, you would use the power of a power rule. If the side length of a cube is a 4 , its volume is ( a 4 ) 3 = a 12 . This rule also applies in financial calculations, such as compound interest, where exponents are used to determine the growth of investments over time. Knowing how to simplify expressions with exponents helps in making these calculations more manageable and understandable.
