apply the product rule of integer exponents to find the numerical equivalent of 12^-15 ⋅ 12^7. Show your work and explain how you used the product rule of integer exponents
To apply the product rule of integer exponents, we need to add the exponents when multiplying two numbers with the same base.
The product rule states:
a^m ⋅ a^n = a^(m + n)
Let's simplify the expression 12^-15 ⋅ 12^7 using the product rule:
12^-15 ⋅ 12^7 = 12^(-15 + 7)
Next, we add the exponents:
-15 + 7 = -8
Therefore, 12^-15 ⋅ 12^7 equals 12^-8.
Explanation:
By applying the product rule of integer exponents, we added the exponents -15 and 7 to get -8. We can now represent 12^-15 ⋅ 12^7 as 12^-8.
To find the numerical equivalent of 12^-15 ⋅ 12^7 using the product rule of integer exponents, we can combine the bases (12) and add the exponents.
The product rule states that when you multiply two numbers with the same base, you add their exponents. In other words:
a^m ⋅ a^n = a^(m + n)
Now, let's apply the product rule to 12^-15 ⋅ 12^7:
12^-15 ⋅ 12^7 = 12^(-15 + 7)
First, let's simplify the exponents:
-15 + 7 = -8
Now, we have:
12^-15 ⋅ 12^7 = 12^(-8)
To compute the numerical equivalent, we can rewrite 12^(-8), using the fact that a negative exponent indicates the reciprocal of the positive exponent:
12^(-8) = 1 / 12^8
Now, let's compute the numerical value of 12^8:
12^8 = 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 = 429,981,696
Therefore, the numerical equivalent of 12^-15 ⋅ 12^7 is:
1 / 12^8 = 1 / 429,981,696 ≈ 2.32523 × 10^(-9)
So, the result is approximately 2.32523 × 10^(-9).
To apply the product rule of integer exponents, we can multiply the two terms by adding their exponents.
Let's break down the expression first:
12^-15 ⋅ 12^7
Using the product rule, we can add the exponents of 12:
(-15 + 7)
Simplifying further, we get:
-8
Therefore, the numerical equivalent of 12^-15 ⋅ 12^7 is 12^(-8).
Here's the step-by-step explanation of how the product rule was used:
1. Start with the given expression: 12^-15 ⋅ 12^7.
2. Apply the product rule by adding the exponents of the two terms: (-15 + 7).
3. Simplify the sum of the exponents: -8.
4. Rewrite the expression with the new exponent as 12^(-8).
5. The numerical equivalent of 12^-15 ⋅ 12^7 is 12^(-8).
Remember, the product rule of integer exponents states that when multiplying two terms with the same base, you can simplify by adding the exponents.