Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4\2^9 ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?

Question

Which of the following is an equivalent expression to (2^−3 ⋅9^3)^−4\2^9  ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?

GPT 3.5 · Answer

To simplify the expression and convert all exponents to positive, we can apply the properties of integer exponents.

We start by simplifying the base of the first parenthesis:

(2^−3 ⋅ 9^3) = (1/2^3 ⋅ 9^3) = (1/8 ⋅ 729) = 729/8

Next, we simplify the base of the second parenthesis:

(2^9 ⋅ 9^−10) = (2^9 ⋅ 1/9^10) = (2^9 ⋅ 1/9^10)

Now, we can rewrite the expression as:

(729/8)^−4 / (2^9 ⋅ 1/9^10) = (729/8)^−4 / (2^9 / 9^10)

To simplify the expression further, we can apply another property of exponents which states that (a/b)^n = a^n / b^n:

(729/8)^−4 / (2^9 / 9^10) = (729^−4 / 8^−4) / (2^9 / 9^10)

Now, we can simplify the numerator and denominator separately:

729^−4 = (3^6)^−4 = 3^(-6*4) = 3^(-24)

8^−4 = (2^3)^−4 = 2^(-3*4) = 2^(-12)

2^9 = 2^(3*3) = (2^3)^3 = 8^3

1/9^10 = (1/3^2)^10 = 3^(-2*10) = 3^(-20)

Substituting these values back into the expression:

(3^(-24) / 2^(-12)) / (8^3 / 3^(-20))

Finally, we use the property of exponents again to rewrite the expression as a single fraction:

(3^(-24) * 3^(-20)) / (2^(-12) * 8^3)

Now, we can simplify the expression by adding the exponents with the same base:

3^(-24-20) / 2^(-12+3) = 3^(-44) / 2^(-9)

Finally, we convert the negative exponents to positive and rewrite the expression as:

2^9 / 3^44

Therefore, the equivalent expression to (2^−3 ⋅9^3)^−4/2^9 ⋅9^−10 with only positive exponents is 2^9 / 3^44.