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^-1/2^9⋅9^−10
2^3 ⋅9^2
2^12⋅9^−12/2^9⋅9^−10
2^3/9^2
Well, let me try to generate a humorous response for you:
Why did the number go to the therapist? Because it had negative exponents and needed a positive outlook!
But on a more serious note, let's simplify that expression. By applying the properties of integer exponents, we can rewrite it as:
(2^-3 * 9^3) - 4^2 * 9^-10
And that simplifies further to:
(1/8 * 729) - 16 * 1/9^10
Which can be written as:
729/8 - 16/9^10
So, out of the given options, the equivalent expression with only positive exponents is not listed.
To simplify the expression and convert the exponents to positive values, we can apply the properties of integer exponents.
Given expression: (2−3 ⋅93)−429 ⋅9−10
Step 1: Apply the exponent rules for multiplication and division.
2^(-3) = 1/2^3
9^3 = (3^2)^3 = 3^(2*3) = 3^6
Rewritten expression: (1/2^3 * 3^6) - 4^2 * 9^(-10)
Step 2: Apply the exponent rule for negative exponents.
9^(-10) = 1/9^10
Rewritten expression: (1/2^3 * 3^6) - 4^2 * (1/9^10)
Step 3: Simplify the expression further.
1/2^3 = 1/8
1/9^10 remains the same
Rewritten expression: (1/8 * 3^6) - 4^2 * (1/9^10)
Step 4: Simplify the exponents.
3^6 = (3^2)^3 = 9^3
Rewritten expression: (1/8 * 9^3) - 4^2 * (1/9^10)
Step 5: Simplify the expression by multiplying and dividing.
1/8 * 9^3 = 9^3/8
4^2 = 16
Final equivalent expression: 9^3/8 - 16 * (1/9^10)
Therefore, the equivalent expression with only positive exponents is 9^3/8 - 16 * (1/9^10).
To find the equivalent expression with only positive exponents, we need to simplify the given expression using the properties of integer exponents.
The given expression is (2−3 ⋅93)−429 ⋅9−10. Let's simplify it step by step:
Step 1: Simplify the exponent of 9^3 to get rid of the negative exponent:
(2−3 ⋅93)−429 ⋅(1/9^3)⋅9−10
= (2−3 ⋅9^3)−429 ⋅1⋅(1/9^3)⋅1⋅(1/9^10)
Step 2: Simplify the exponent of 9^3 and 9^10:
= (2−3 ⋅9^3)−429 ⋅1⋅(1/(9^3)⋅1/(9^10))
= (2−3 ⋅9^3)−429 ⋅(1/(9^3+10))
= (2−3 ⋅9^3)−429 ⋅(1/9^13)
Step 3: Simplify the exponent of 9^3:
= (2−3 ⋅27)−429 ⋅(1/9^13)
= (2−81)−429 ⋅(1/9^13)
= (−79)−429 ⋅(1/9^13)
So, the simplified expression with only positive exponents is:
2^79/9^13
Therefore, the correct answer is not provided in the given options.
To rewrite the expression with positive exponents, we can apply the property of negative exponents, where a^(-n) = 1/a^n.
Starting from the original expression:
(2-3 * 9^3)-4^2 * 9^-10
First, apply the property of negative exponents to the 3 in the first term:
(2-3 * (1/9^3))-4^2 * 9^-10
Next, simplify the exponent of 9 in the first term and distribute the exponent of -10 to both the 4 and 9 in the second term:
(2-3 * 1/729)-4^2 * (1/9^10)
Now, simplify the numerical values in the first term:
(2-1/243)-4^2 * (1/9^10)
Next, simplify the exponent of 4 in the second term:
(2-1/243)-16 * (1/9^10)
Now, simplify the numerical values in the second term:
(2-1/243)-16/9^10
Finally, rewrite the exponents with positive values:
2^1-1/3^5-16/3^20
So, the equivalent expression with only positive exponents is 2^1-1/3^5-16/3^20.