which expression is correctly developed to use the product rule of exponents? (5^2) ^9, 32^7, 10^8 x 10^8, 6^2 x 7^3
The expression that is correctly developed to use the product rule of exponents is 10^8 x 10^8. The product rule of exponents states that when multiplying two numbers with the same base, you add the exponents. In this case, both exponents are 8, so the expression simplifies to 10^(8+8) = 10^16.
To use the product rule of exponents, we can apply the rule that states:
(a^m) x (a^n) = a^(m + n)
Let's evaluate each expression using this rule:
1. (5^2) ^9
Applying the rule, we get:
(5^2) ^9 = 5^(2 x 9) = 5^18
2. 32^7
There is no need to apply the product rule of exponents in this case because we are only dealing with a single base and exponent. Therefore, 32^7 is the correct and simplified expression.
3. 10^8 x 10^8
Applying the product rule of exponents, we get:
10^8 x 10^8 = 10^(8 + 8) = 10^16
4. 6^2 x 7^3
Applying the product rule of exponents, we get:
6^2 x 7^3 = 6^(2 + 3) x 7^3 = 6^5 x 7^3
Therefore, the correct expression to use the product rule of exponents is:
(5^2) ^9, 32^7, 10^8 x 10^8, 6^2 x 7^3
To correctly use the product rule of exponents, we need to remember that when multiplying two exponential expressions with the same base, we keep the base the same and add the exponents.
Let's analyze each expression and decide which one uses the product rule correctly:
1. (5^2) ^9: To solve this expression, we need to simplify the exponent first. (5^2) is equal to 25, so the expression becomes 25^9. There is no multiplication involved, so the product rule is not needed here.
2. 32^7: This expression does not involve exponentiation with the same base being multiplied together, so the product rule is not applicable here.
3. 10^8 x 10^8: This expression involves the product of two exponential expressions with the same base, which is 10. To apply the product rule correctly, we keep the base the same and add the exponents: 10^(8+8) = 10^16.
4. 6^2 x 7^3: Similar to the previous expression, this one involves multiplying two exponential expressions with the same base. To apply the product rule, we keep the base the same and add the exponents: 6^(2+3) x 7^3 = 6^5 x 7^3.
In summary, the expression that is correctly developed to use the product rule of exponents is 10^8 x 10^8.