Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

(43 ⋅ 7−2)−579 ⋅ 4−13=

Answer 1

[AI-generated answer — Bot GPT 3.5]

To apply the properties of integer exponents, we need to use the following rules:

1. Any number raised to the power of 0 is equal to 1.
2. The product of two numbers with the same base raised to different exponents is equal to the base raised to the sum of the exponents.
3. The quotient of two numbers with the same base raised to different exponents is equal to the base raised to the difference of the exponents.

Let's apply these rules to the expression and generate an equivalent expression with only positive exponents:

(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)

Using rule 2, we can rewrite 7^-2 as 1/7^2 and 4^-13 as 1/4^13:

(43 ⋅ 1/7^2)^(-579) ⋅ 1/4^13

Simplifying the exponents:

(43 ⋅ 1/49)^(-579) ⋅ 1/4^13

Using rule 3, we can rewrite (43 ⋅ 1/49)^(-579) as (43)^(-579) ⋅ (1/49)^(-579):

(43)^(-579) ⋅ (1/49)^(-579) ⋅ 1/4^13

Applying rule 1 to (1/49)^(-579), we can rewrite it as 49^579:

(43)^(-579) ⋅ 49^579 ⋅ 1/4^13

Since 1/4^13 has a negative exponent, we can rewrite it as 4^-13 using rule 2:

(43)^(-579) ⋅ 49^579 ⋅ 4^-13

Now, let's solve the expression:

The positive exponents are (43)^(-579) and 49^579. To simplify them, we can rewrite them using reciprocals:

(1/43)^579 and (1/49)^579

Since the bases are reciprocals, we can simplify them further:

1/(43^579) and 1/(49^579)

Finally, multiplying all the terms together:

(1/(43^579)) ⋅ (1/(49^579)) ⋅ (1/4^13)

The expression is simplified and cannot be simplified further.

Therefore, the equivalent expression with only positive exponents is:

(1/(43^579)) ⋅ (1/(49^579)) ⋅ (1/4^13)