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)
To use the product rule of integer exponents, we need to apply it to each term separately.
Starting with the first term, 12^(-5), we can rewrite it as 1/(12^5). This is because a negative exponent means to take the reciprocal of the base raised to the positive exponent.
Next, we have 127. We can rewrite it as 127^1.
Now, we can use the product rule of integer exponents, which states that when multiplying two terms with the same base raised to different exponents, we can simply add the exponents.
So, 12^(-5) * 127^1 becomes 1/(12^5) * 127^1.
Applying the product rule, we add the exponents: 1/(12^5 * 127^1).
Now, we can simplify the expression by evaluating the base raised to the exponents. 12^5 = 248,832 and 127^1 = 127.
Therefore, the numerical equivalent of 12^(-5) * 127^1 is 1/(248,832 * 127) = 1/31,604,064.
The product rule of integer exponents states that for any real numbers a and b and any integer n, (ab)^n = a^n * b^n.
In this case, we are asked to find the numerical equivalent of 12^(-5) * 127.
Using the product rule of integer exponents, we can rewrite this expression as (12 * 127)^(-5).
Next, we calculate the numerical value of 12 * 127:
12 * 127 = 1,524.
Now we have (1,524)^(-5).
To find the numerical equivalent, we apply the power rule for negative exponents, which states that for any nonzero real number a, a^(-n) = 1 / (a^n).
Therefore, (1,524)^(-5) = 1 / (1,524)^5.
Calculating (1,524)^5, we get 1,524 * 1,524 * 1,524 * 1,524 * 1,524 = 12,330,877,596,176.
Finally, we substitute this value back into the expression:
1 / (12,330,877,596,176) = 8.12 * 10^(-14).
So, the numerical equivalent of 12^(-5) * 127 is approximately 8.12 * 10^(-14).
To apply the product rule of integer exponents, we need to remember the formula:
a^m * a^n = a^(m + n)
In this case, we have 12^(-5) * 127. According to the product rule, we add the exponents.
First, let's calculate 12^(-5):
12^(-5) = 1 / (12^5)
Now, let's calculate 12^5:
12^5 = 12 * 12 * 12 * 12 * 12 = 248,832
So, 12^(-5) = 1 / 248,832
Now, let's multiply 12^(-5) with 127:
1 / 248,832 * 127 = 127 / 248,832
Therefore, the numerical equivalent of 12^(-5) * 127 is 127 / 248,832.
We used the product rule of integer exponents to break down the expression and simplify it by combining the exponents. This rule allows us to calculate the result more efficiently by manipulating the powers of the base.