Apply the product rule of integer exponents to find the numerical equivalent of 12^−5⋅12^7 . Show your work and explain how you used the product rule of integer exponents. (2 points)
To find the numerical equivalent of 12^(-5) * 12^7 using the product rule of integer exponents, we need to multiply the base (12) and add the exponents (-5 and 7) together.
Using the product rule, we add the exponents: -5 + 7 = 2.
So, we have 12^(-5) * 12^7 = 12^2.
Now, we can evaluate 12^2, which is equal to 144.
Therefore, the numerical equivalent of 12^(-5) * 12^7 is 144.
To apply the product rule of integer exponents, we add the exponents together. The product rule states that for any positive integers a and b and any integer n:
(a^n)⋅(a^m) = a^(n+m)
In this case, we have 12^(-5)⋅12^7.
First, let's simplify 12^(-5). According to the rule for negative exponents, any non-zero number raised to the power of -n is equal to 1 divided by that number raised to the power of n. So:
12^(-5) = 1/(12^5)
Next, we apply the product rule by combining the exponents:
12^(-5)⋅12^7 = 1/(12^5)⋅12^7
To simplify further, we can divide 12^7 by 12^5:
12^(-5)⋅12^7 = 1/(12^5)⋅(12^2⋅12^5)
Using the product rule again:
12^(-5)⋅12^7 = 1/(12^5)⋅12^(2+5)
Simplifying the exponents:
12^(-5)⋅12^7 = 1/(12^5)⋅12^7
Now we multiply the coefficients:
12^(-5)⋅12^7 = 12^(7-5) = 12^2
Therefore, the numerical equivalent of 12^(-5)⋅12^7 is 12^2 or 144.
To apply the product rule of integer exponents, we need to know the rule first. The product rule states that when multiplying two terms with the same base, we can add their exponents together.
In this case, we have 12^(-5) * 12^7. Both terms have the base of 12, so we can apply the product rule by adding the exponents:
12^(-5) * 12^7 = 12^(-5+7) = 12^2
To simplify further, let's calculate 12^2:
12^2 = 12 * 12 = 144
Therefore, the numerical equivalent of 12^(-5) * 12^7 is 144.