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
2392
Start Fraction 2 cubed over 9 squared end fraction
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
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
The equivalent expression is 2^3 * 9^2, which is option (b).
To find an equivalent expression with only positive exponents, we need to simplify the given expression using the properties of integer exponents:
(2−3 ⋅ 9^3)−4^2 ⋅ 9^−10
To simplify this expression, we can follow these steps:
Step 1: Simplify any negative exponents by moving them to the denominator:
(2^−3 ⋅ (9^3)^−1) ⋅ (4^2 ⋅ (9^−10)^1)
Step 2: Evaluate the exponent of negative powers. Raise each base to the power of the exponent:
(2^−3 ⋅ 9^−3) ⋅ (4^2 ⋅ 9^−10)
Step 3: Simplify the expression further by multiplying the coefficients and combining like terms:
2^−3 ⋅ 4^2 ⋅ 9^−3 ⋅ 9^−10
Step 4: Apply the property of multiplying exponents with the same base, and add the exponents:
2^−3 ⋅ 4^2 ⋅ 9^(−3−10)
Step 5: Simplify the expression by evaluating the exponents and performing the calculations:
2^−3 = 1/2^3 = 1/8
4^2 = 16
9^−3 = 1/9^3 = 1/729
9^−10 = 1/9^10
Substituting these values into the expression, we get:
(1/8) ⋅ 16 ⋅ (1/729) ⋅ (1/9^10)
Step 6: Multiply the coefficients and simplify:
(1/8) ⋅ 16 ⋅ (1/729) ⋅ (1/9^10) = 2/729 ⋅ (1/9^10)
Step 7: Rewrite 9^10 with only positive exponents:
2/729 ⋅ 1/(1/9^10) = 2/729 ⋅ 9^10
Therefore, the equivalent expression with only positive exponents is 2/729 ⋅ 9^10.
To find the equivalent expression with only positive exponents, we need to apply the properties of integer exponents.
First, let's simplify the expression (2−3 ⋅93)−429 ⋅9−10 step by step:
1. Simplify the exponent: (2−3 ⋅9^3)−429 ⋅9^−10
2. Apply the power of a product rule: (2−3 ⋅9^3−429) ⋅9^−10
3. Simplify the exponent: (2^−3 ⋅9^3−429) ⋅9^−10
4. Apply the power of a product rule: (2^−3 ⋅9^3)^-429 ⋅9^−10
5. Apply the power of a power rule: (2^−3 ⋅(9^3)^-429) ⋅9^−10
6. Simplify the exponents: (2^−3 ⋅9^-1287) ⋅9^−10
7. Apply the quotient rule: 2^(-3 + -1287) ⋅9^(-10 + -1287)
8. Simplify the exponents: 2^-1290 ⋅9^-1297
Now, let's look at the given options:
1. 2392: This option does not match the expression we simplified.
2. Start Fraction 2 cubed over 9 squared end fraction: This is equivalent to 2^3 / 9^2. It does not match the expression we simplified.
3. 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: This option does not match the expression we simplified.
4. 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: This option does not match the expression we simplified.
5. 23 ⋅ 92: This option is equal to 2^3 ⋅ 9^2. It matches the expression we simplified.
Therefore, the equivalent expression with only positive exponents is 23 ⋅ 92.