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)
The product rule of integer exponents states that when you multiply two exponential expressions with the same base, you add the exponents.
In this case, we have 12 ^ -5 * 12^7. These expressions have the same base, which is 12.
Using the product rule, we add the exponents:
12^(-5+7)
Now we simplify the exponent:
12^2
Therefore, the numerical equivalent of 12 ^ - 5 * 12 ^ 7 is 12^2.
To apply the product rule of integer exponents, we use the formula:
a^m * a^n = a^(m+n)
In this case, we have 12^(-5) * 12^7. We can rewrite the expression using the product rule as 12^(-5 + 7).
Now, let's simplify the exponent:
-5 + 7 = 2
Therefore, 12^(-5 + 7) becomes 12^2.
To calculate the numerical equivalent, we square 12, which gives us:
12^2 = 144
Hence, the numerical equivalent of 12^(-5) * 12^7 is 144.
To find the numerical equivalent of 12^-5 * 12^7 using the product rule of integer exponents, we need to apply the rule which states that when multiplying two numbers with the same base, we add their exponents.
Let's break down the steps:
Step 1: Simplify the base.
12 can be written as 2^2 * 3.
Step 2: Substitute the base with the simplified form.
(2^2 * 3)^-5 * (2^2 * 3)^7
Step 3: Apply the product rule by adding the exponents.
2^(-5 * 2) * 3^(-5) * 2^(7 * 2) * 3^(7)
Step 4: Simplify the exponents.
2^(-10) * 3^(-5) * 2^14 * 3^7
Step 5: Add the powers of the same base.
2^(-10 + 14) * 3^(-5 + 7)
Step 6: Simplify the exponents.
2^4 * 3^2
Step 7: Evaluate the result.
2^4 = 2 * 2 * 2 * 2 = 16
3^2 = 3 * 3 = 9
Therefore, the numerical equivalent of 12^-5 * 12^7 is 16 * 9 = 144.