Apply the product rule of integer exponents to find the numerical equivalent of 12−⁵• 12⁷ . Show your work and explain how you used the product rule of integer exponents.(2 points)
The product rule of integer exponents states that when you have two numbers with the same base and different exponents, you can multiply the bases and add the exponents. In this case, we have 12 raised to the power of -5 (12^(-5)) multiplied by 12 raised to the power of 7 (12^7).
To find the numerical equivalent, we can first simplify each exponent separately.
12^(-5) can be rewritten as 1/(12^5), which means we need to find the reciprocal of 12^5.
12^7 can be calculated by multiplying 12 seven times: 12 * 12 * 12 * 12 * 12 * 12 * 12.
Now, we can multiply these two numbers together:
(1/(12^5)) * (12^7)
Since the bases are the same (12), we can use the product rule of integer exponents and add the exponents:
1/(12^5 * 12^(-7))
Next, we can simplify the expression in the denominator using the product rule of integer exponents:
1/12^(5 + (-7))
Simplifying further, we have:
1/12^(-2)
With the negative exponent in the denominator, we can move it to the numerator by changing its sign:
12^2
Finally, we can calculate the numerical value of 12^2:
12^2 = 12 * 12 = 144
Therefore, the numerical equivalent of 12^(-5) * 12^7 is 144.