Write the expression [tex]a^{15}[/tex] as a product of two exponents with the same base, one of which equals [tex]a^2[/tex]
Answer (1)
We are given a 15 and need to express it as a 2 ". a x .
Using the exponent rule a m ". a n = a m + n , we get a 15 = a 2 + x .
Equating the exponents, we have 15 = 2 + x .
Solving for x , we find x = 13 , so a 15 = a 2 ⋅ a 13 . The final answer is a 2 ". a 13 .
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, where one of the exponents is a 2 . This means we want to find an exponent x such that a 15 = a 2 ⋅ a x .
Applying Exponent Rules Using the property of exponents that states a m ⋅ a n = a m + n , we can rewrite the equation as a 15 = a 2 + x .
Equating Exponents Since the bases are equal, we can equate the exponents: 15 = 2 + x .
Solving for x Solving for x , we subtract 2 from both sides of the equation: x = 15 − 2 = 13 .
Final Answer Therefore, a 15 = a 2 ⋅ a 13 . The expression a 15 as a product of two exponents with the same base, one of which equals a 2 is a 2 ⋅ a 13 .
Examples
Understanding how to manipulate exponents is crucial in many areas, such as calculating compound interest. For example, if you invest money and it grows at a certain percentage each year, the exponent represents the number of years the money is invested. By understanding how to break down and combine exponents, you can easily calculate the future value of your investment or compare different investment options.
