Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7^−3)^7 ⋅ 7^20/6^−2=

Answer 1

To simplify the expression and eliminate negative exponents, we can start by applying the property of exponentiation, which states that (a^m)^n = a^(m*n).

So, we can rewrite (7^(-3))^7 as 7^(-3*7), which simplifies to 7^(-21).

Next, we can use the property of exponentiation that states a^(-n) = 1/(a^n). Applying this property to 7^(-21), we get 1/(7^21).

Similarly, we can apply the property of exponentiation to 7^20, giving us 7^(20*(1/6)), which simplifies to 7^(20/6).

We can also apply the property of exponentiation to 6^(-2), resulting in 1/(6^2).

Now, we can rewrite the expression as (1/(7^21))*(7^(20/6))/(1/(6^2)).

To simplify further, we can multiply the numerator and denominator by (6^2), which gives us (1/(7^21))*(7^(20/6))*(6^2).

Now, let's simplify each term:

1/(7^21) cannot be simplified further.

7^(20/6) can be simplified as (7^(2/6))^10. The (2/6) simplifies to (1/3). So, we are left with (7^(1/3))^10.

Applying the property of exponentiation again, we get (7^(1/3))^10 = 7^(10*(1/3)), which simplifies to 7^(10/3).

Finally, we can multiply all the terms together:

(1/(7^21))*(7^(20/6))*(6^2) = (1/(7^21))*(7^(10/3))*(6^2).

This expression cannot be simplified further, so the final answer is:

(1/(7^21))*(7^(10/3))*(6^2) as an improper fraction.