Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses
2^−7⋅9^-1/2^9⋅9^−10
2^3 ⋅9^2
2^12⋅9^−12/2^9⋅9^−10
2^3/9^2
The correct answer is 2^3/9^2.
To find the equivalent expression with only positive exponents, we can use the properties of exponents.
Starting with the given expression:
(2−3 ⋅93)−429 ⋅9−10
First, let's simplify the exponent with negative powers:
2^(-3) = 1/2^3 = 1/8
(1/8 ⋅9^3)−429 ⋅1/9^10
Next, let's simplify the exponent with fractional powers:
9^3 = (3^2)^3 = 3^(2*3) = 3^6
(1/8 ⋅3^6)−429 ⋅1/9^10
Now, let's simplify the exponents with negative powers again:
1/9^10 = 9^(-10) = (3^2)^(-10) = 3^(-20)
(1/8 ⋅3^6)−429 ⋅3^(-20)
Now, let's combine the fractions:
1/8 ⋅3^6 = (1 ⋅ 3^6) / 8 = 3^6 / 8
(3^6 / 8)−429 ⋅3^(-20)
Finally, let's simplify the expression:
(3^6 / 8)−429 ⋅3^(-20) = 3^6 / 8 ⋅ 3^(-20) = 3^6 ⋅ 3^(-20) / 8 = 3^(6-20) / 8 = 3^(-14) / 8
Therefore, the equivalent expression with only positive exponents is:
3^(-14) / 8