Question

Apply the properties of integer exponents to generate an equivalent expression to (7^-3)^7 x 7^20/6^-2 with only positive exponents then solve the expression. Provide the answer as an improper fraction

Answer 1

To apply the properties of integer exponents, we need to simplify each expression separately and then combine them.

First, let's simplify (7^-3)^7:
Using the property (a^m)^n = a^(m*n), we can rewrite it as 7^(-3*7) = 7^-21.

Next, let's simplify 7^20:
Since we need to convert it to a positive exponent, we can write it as (7^20/1) = (7^20/1^6) using the property a^0 = 1. Then, applying the exponent property (a^m)^n = a^(m*n), we get 7^(20*1/6) = 7^(20/6) = 7^(10/3).

Now, let's simplify 6^-2:
Using the property a^(-m) = 1/(a^m), we get 6^-2 = 1/(6^2) = 1/36.

Combining the simplified expressions, we have:
(7^-3)^7 x 7^20/6^-2 = 7^-21 x 7^(10/3) x (1/36).

To multiply expressions with the same base, we can add their exponents, so we have:
7^(-21 + 10/3) x (1/36).

To rewrite the exponent as an improper fraction, we have (10/3) = 3 1/3 = 10/3.

Now, we can combine the fractions in the exponent:
7^(10/3) = 7^(3 1/3) = 7^(3 + 1/3) = 7^3 x 7^(1/3) = 343 x (7^(1/3)).

Therefore, the equivalent expression with only positive exponents is:
(7^-3)^7 x 7^20/6^-2 = 343 x (7^(1/3)) x (1/36).

Now, we can evaluate the expression:
343 x (7^(1/3)) x (1/36) = 343 x (7^(1/3))/36 = 343/36 x (7^(1/3)).

So, the answer as an improper fraction is 343/36 x (7^(1/3)).