To simplify this expression and have only positive exponents, we can use the following properties of integer exponents:
1) When a negative exponent appears in the numerator, it can be moved to the denominator and become positive exponent:
a^(-n) = 1/a^n
2) When a negative exponent appears in the denominator, it can be moved to the numerator and become positive exponent:
1/a^(-n) = a^n
Using these properties, we can simplify the expression as follows:
(2^(-3) ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10)
= (1/(2^3) ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10) (Moving negative exponents from numerator to denominator)
= (1/8 ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10) (Evaluating 2^3 and simplifying)
= (9^3 / 8)^(-4) / 2^9 ⋅ 9^(-10) (Applying properties of exponents to the numerator)
= (9^3)^(-4) / 8^(-4) / 2^9 ⋅ 9^(-10) (Simplifying the expression inside parentheses)
= (1 / 9^(3*4)) / 1 / (8^4) / 2^9 ⋅ 9^(-10) (Applying properties of exponents to the numerator and denominator)
= 1 / 9^12 / 1 / (8^4) / 2^9 ⋅ 9^(-10) (Simplifying exponents)
= 2^9 ⋅ 9^(-10) / 9^12 / 8^4 (Moving negative exponent to numerator)
= 2^9 / 8^4 ⋅ 9^(-10) / 9^12 (Simplifying fractions)
= 2^9 / 8^4 ⋅ 1/ (9^10 ⋅ 9^12) (Simplifying exponents)
= 2^9 / 8^4 ⋅ 1 / 9^(10+12) (Applying properties of exponents)
= 2^(9-4) / 2^(9*4) ⋅ 1 / 9^22 (Simplifying exponents)
= 2^5 / 2^36 ⋅ 1 / 9^22 (Simplifying exponents)
= 1 / 2^(36-5) ⋅ 1 / 9^22 (Applying properties of exponents)
= 1 / 2^31 ⋅ 1/ 9^22 (Simplifying exponents)
Therefore, the equivalent expression with only positive exponents is 1 / 2^31 ⋅ 1/ 9^22.
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
2^−7⋅9^−1/2^9⋅9^−10
2^3 ⋅9^2
2^12⋅9^-12/2^9⋅9^−10
2^3/9^2
1 answer