Apply the Properties of Integer Exponents to generate equivalent expressions to
with only positive exponents, then solve the expression. Provide the answer as an improper fraction. (1 point)
(7^-3)^7x7^20/6^-2
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression inside the parentheses:
(7^-3)^7 = 7^(-3*7) = 7^-21
Now let's simplify the expression in the numerator:
7^20 = 7^(2*10) = (7^2)^10 = 49^10
And simplify the expression in the denominator:
6^-2 = 1/6^2 = 1/36
Now let's rewrite the expression using the simplified forms:
(7^-3)^7 * 7^20 / 6^-2 = 7^-21 * 49^10 / 1/36
To divide by a fraction, we can multiply by its reciprocal:
7^-21 * 49^10 * 36
Let's now simplify this expression:
7^-21 * 49^10 * 36 = (1/7^21) * 49^10 * 36 = 49^10 * 36 / 7^21
To simplify further, we can express 49 as a power of 7:
49^10 = (7^2)^10 = 7^20
Our expression becomes:
7^20 * 36 / 7^21
Since we're dividing with the same base, we can subtract the exponents:
7^20 * 36 / 7^21 = (7^20 / 7^21) * 36 = 7^(20-21) * 36 = 7^-1 * 36
Finally, let's simplify 7^-1:
7^-1 = 1/7^1 = 1/7
Our expression is now:
1/7 * 36 = 36/7
The answer as an improper fraction is 36/7.