Write the expression $a^{15}$ as a product of two exponents with the same base, one of which equals $a^6$.
Answer (1)
We are given a 15 and want to express it as a 6 "." a x .
Using the property of exponents, we have a 15 = a 6 + x .
Equating the exponents, we get 15 = 6 + x .
Solving for x , we find x = 9 , so a 15 = a 6 ⋅ a 9 .
Explanation
Understanding the Problem We are given the expression a 15 and we want to express it as a product of two exponents with the same base a , where one of the exponents is a 6 .
Setting up the Equation Let the other exponent be a x . Then we can write the equation: a 15 = a 6 ⋅ a x
Applying Exponent Rules Using the property of exponents, when multiplying exponents with the same base, we add the exponents. So, we have: a 15 = a 6 + x
Equating Exponents Since the bases are the same, we can equate the exponents: 15 = 6 + x
Solving for x Now, we solve for x by subtracting 6 from both sides of the equation: x = 15 − 6 x = 9
Final Answer Therefore, the expression a 15 can be written as the product of a 6 and a 9 :
a 15 = a 6 ⋅ a 9
Examples
Understanding exponent rules is crucial in various fields, such as calculating compound interest. For instance, if you invest P dollars at an annual interest rate r compounded n times per year for t years, the future value A of the investment is given by A = P ( 1 + n r ) n t . If you want to analyze how the investment grows over time, you need to understand how to manipulate exponents. Simplifying expressions with exponents helps in predicting the long-term growth of your investment and making informed financial decisions.
