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⋅9−129⋅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
23 ⋅92
2 cubed times 9 squared
2392
Start Fraction 2 cubed over 9 squared end fraction
212⋅9−1229⋅9−10
The equivalent expression with only positive exponents is:
2^(-7) * (9^(-1) / (2^9 * 9^(-10)))
Simplifying this expression further:
= 2^(-7) * (9^(-1) / 2^9) * 9^(-10)
Therefore, the correct option is:
2^(-7) * 9^(-1) * 2^(-9) * 9^(-10)
To find an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
Let's break down step by step:
Step 1: Simplify the exponents in (2−3⋅9^3)−4^2⋅9^−10:
2−3⋅9^3 simplifies to 2−3⋅729
4^2⋅9^−10 simplifies to 16⋅9^−10
Step 2: Apply the property of negative exponents:
2−3⋅729 can be rewritten as 2−3/729 or 2−3/(9^3)
16⋅9^−10 can be rewritten as 16/9^10
Step 3: Combine the expressions:
(2−3/729)−(16/9^10)
Step 4: Simplify the expression:
To simplify further, we need to find a common denominator for the fractions. The common denominator will be 729.
(2−3/729) can be rewritten as (2*729−3)/(729)
(16/9^10) can be rewritten as (16*9^10)/(9^10*729)
Now we can combine the fractions using a common denominator:
(2*729−3)/(729)−(16*9^10)/(9^10*729)
Step 5: Simplify the expression further by simplifying the numerator and denominator:
(1456−3)/(729)−(144*9^10)/(9^10*729)
Step 6: Combine like terms in the numerator:
(1453)/(729)−(144*9^10)/(9^10*729)
Step 7: Simplify the expression by canceling out common factors:
(1453)/(729)−(144)/(729)
Step 8: Combine the fractions:
(1453−144)/(729)
Step 9: Simplify the numerator:
(1309)/(729)
Therefore, the equivalent expression with only positive exponents is 1309/729.