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
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.
To simplify the given expression and convert all exponents to positive, we can apply the properties of integer exponents.
Let's break down the expression step-by-step:
Step 1: Simplify the exponents within parentheses:
(2^-3 ⋅ 9^3)^-4 = 2^(-3 * -4) ⋅ 9^(3 * -4) = 2^12 ⋅ 9^-12
Step 2: Combine the two fractions by subtracting the exponents:
2^12 ⋅ 9^-12 / 2^9 ⋅ 9^-10
Step 3: Apply the properties of exponents to get rid of the negative exponents:
2^(12 - 9) ⋅ 9^(-12 - (-10)) = 2^3 ⋅ 9^(-2)
So, the equivalent expression with only positive exponents is:
2^3 ⋅ 9^2
To simplify the expression and generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's break it down step by step:
Step 1: Simplify the exponents within parentheses:
(2^−3 ⋅ 9^3)^−4 becomes (1/2^3 ⋅ 9^3)^−4.
Step 2: Apply the property of negative exponents:
(1/2^3 ⋅ 9^3)^−4 can be written as (9^3/2^3)^−4.
Step 3: Simplify the exponents outside parentheses:
(9^3/2^3)^−4/2^9 ⋅ 9^−10 becomes (9^3/2^3)^−4/2^9 ⋅ (1/9^10).
Step 4: Apply the property of exponent multiplication:
(9^3/2^3)^−4/2^9 ⋅ (1/9^10) can be written as (9^3/2^3)^−4 ⋅ (1/2^9 ⋅ 9^10).
Step 5: Apply the property of exponent division:
(9^3/2^3)^−4 ⋅ (1/2^9 ⋅ 9^10) can be written as (9^3)^−4/(2^3)^−4 ⋅ (1/2^9 ⋅ 9^10).
Step 6: Simplify the exponents within parentheses:
(9^3)^−4/(2^3)^−4 ⋅ (1/2^9 ⋅ 9^10) becomes 9^−12/2^−12 ⋅ (1/2^9 ⋅ 9^10).
Step 7: Apply the property of negative exponents:
9^−12/2^−12 ⋅ (1/2^9 ⋅ 9^10) can be written as (1/9^12)/(1/2^12) ⋅ (1/2^9 ⋅ 9^10).
Step 8: Simplify the exponents outside the parentheses:
(1/9^12)/(1/2^12) ⋅ (1/2^9 ⋅ 9^10) becomes (1/9^12)/(1/2^12) ⋅ 1/2^9 ⋅ 9^10.
Step 9: Apply the property of dividing by a fraction:
(1/9^12)/(1/2^12) ⋅ 1/2^9 ⋅ 9^10 becomes (1/9^12) ⋅ (2^12/1) ⋅ 1/2^9 ⋅ 9^10.
Step 10: Simplify:
(1/9^12) ⋅ (2^12/1) ⋅ 1/2^9 ⋅ 9^10 becomes (2^12⋅9^−12)/(2^9⋅9^10).
Ultimately, the equivalent expression with only positive exponents is:
(2^12⋅9^−12)/(2^9⋅9^10).