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
212⋅9−1229⋅9−10
Start fraction 2 superscript 12 baseline times 9 superscript negative 12 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction
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 2^12 * 9^-12 / 2^9 * 9^-10 is an equivalent expression to (2-3 * 9^3)-4^29 * 9^(-10) with only positive exponents.
To simplify the expression (2−3 * 93)−429 * 9−10 with only positive exponents, we need to apply the properties of integer exponents.
First, let's simplify the exponents:
2−3 = 1 / 2^3 = 1/8
93 = 9^3 = 729
429 = 4^2 * 9 = 16 * 9 = 144
9−10 = 1 / 9^10
Now let's substitute these values back into the expression:
(1/8 * 729)−144 * 1 / 9^10
The next step is to simplify the multiplication and division within the expression:
(729/8) - (144 / 9^10)
Since 144 is divisible by 9, we can simplify further:
(729/8) - (16 / 9^9)
To combine the fractions, we need to find a common denominator:
The common denominator is 8 * 9^9, so we need to multiply each fraction accordingly:
(729 * 9^9) / (8 * 9^9) - (16 * 8) / (9^9 * 8)
Simplifying further, we have:
(729 * 9^9 - 16 * 8) / (8 * 9^9)
At this point, we have simplified the expression as much as possible using the properties of integer exponents. Thus, the equivalent expression with only positive exponents is:
(729 * 9^9 - 16 * 8) / (8 * 9^9)
To find the equivalent expression with positive exponents, we can use the rules of integer exponents.
First, let's simplify (2−3 ⋅ 9^3)−4 ⋅ 9−10 step by step:
1. Simplify 2−3 = 1/(2^3) = 1/8
2. Simplify 9^3 = 729
3. Simplify 9−10 = 1/(9^10) = 1/3486784401
Now, we have (1/8 ⋅ 729)−4 ⋅ (1/3486784401):
1. Multiply 1/8 by 729: (1/8) * 729 = 91.125
2. Simplify 1/3486784401 = (1/9^10)
3. Now, we have (91.125)−4 * (1/9^10)
To express this with positive exponents:
1. Rewrite 91.125 as (9.1125 * 10^1), where 10 is raised to the power of 1.
2. Rewrite 1/9^10 as 9^(-10).
Now, we have (9.1125 * 10^1)−4 * 9^(-10):
1. Distribute the exponent -4 to both terms inside the parentheses:
(9.1125)^(-4) * (10^1)^(-4) * 9^(-10) = (9.1125)^(-4) * 10^(-4) * 9^(-10)
Finally, we can write this in an equivalent form with positive exponents:
(9.1125)^(-4) * 10^(-4) * 9^(-10) = 9^(-4) * 9.1125^(-4) * 10^(-4) * 9^(-10).
So, the equivalent expression with positive exponents is 9^(-4) * 9.1125^(-4) * 10^(-4) * 9^(-10).