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?
Responses
23 ⋅92
2 cubed times 9 squared
2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
2392
Start Fraction 2 cubed over 9 squared end fraction
2^12⋅9^−12/2^9⋅9^−10
The correct equivalent expression with only positive exponents is 23 ⋅ 92.
SO ITS NOT C????
No, it is not option C. The correct equivalent expression is option A, 23 ⋅ 92.
To simplify the expression and convert all the exponents to positive, we can use the properties of integer exponents.
Starting with the expression (2^(-3) * 9^3)^(-4) / 2^9 * 9^(-10), let's simplify step-by-step:
Step 1: Simplify the exponents within the parentheses.
(2^(-3) * 9^3)^(-4) = 2^((-3) * (-4)) * 9^(3 * (-4))
= 2^12 * 9^(-12)
Step 2: Rewrite the division as multiplication using negative exponents.
(2^12 * 9^(-12))/ 2^9 * 9^(-10) = 2^12 * 9^(-12) * 2^(-9) * 9^(-10)
Step 3: Apply the property of multiplying with like bases and adding the exponents.
2^12 * 2^(-9) = 2^(12 - 9) = 2^3
Step 4: Simplify the remaining terms.
2^3 * 9^(-12) * 9^(-10) = 2^3 * 9^((-12) + (-10))
= 2^3 * 9^(-22)
Therefore, the equivalent expression with only positive exponents is 2^3 * 9^(-22).