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?(1 point)
To simplify the expression and convert all the exponents to positive, we can use the properties of integer exponents.
First, let's simplify the numerator: (2^(-3) ⋅ 9^3)^(-4)
We can apply the property (a^m)^n = a^(m*n) to simplify this expression:
(2^(-3) ⋅ 9^3)^(-4) = 2^((-3)*(-4)) ⋅ 9^(3*(-4))
= 2^(12) ⋅ 9^(-12)
Next, let's simplify the denominator: 2^9 ⋅ 9^(-10)
Now, we can apply the property a^(-n) = 1/(a^n) to convert the negative exponent to positive:
9^(-10) = 1 / 9^10
Finally, the expression becomes:
2^12 ⋅ 9^(-12) / (2^9 ⋅ 1 / 9^10)
To simplify further, we can use the property a / (b/c) = a * (c/b):
2^12 ⋅ 9^(-12) / (2^9 ⋅ 1 / 9^10) = 2^12 ⋅ 9^(-12) * (9^10 / 2^9)
Now we can simplify further by multiplying the numbers with the same base:
2^12 ⋅ 9^(-12) * (9^10 / 2^9) = (2^12 / 2^9) ⋅ (9^10 ⋅ 9^(-12))
We can apply the property a^n / a^m = a^(n-m) to simplify further:
(2^12 / 2^9) ⋅ (9^10 ⋅ 9^(-12)) = 2^(12-9) ⋅ 9^(10-12)
= 2^3 ⋅ 9^(-2)
= 8 ⋅ 1 / 9^2
= 8 / 9^2
= 8 / 81
Therefore, the equivalent expression with only positive exponents is 8 / 81.
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?
To simplify the given expression and convert all the exponents to positive, let's begin by applying the properties of integer exponents step by step.
Starting with the numerator expression: (2^(-3) ⋅ 9^3)^(-4)
Using the property (a^m)^n = a^(m*n), we can simplify it as follows:
= 2^((-3)*(-4)) ⋅ 9^(3*(-4))
= 2^(12) ⋅ 9^(-12)
Now, we can simplify the denominator expression: 2^9 ⋅ 9^(-10)
So far, we have:
(2^(12) ⋅ 9^(-12)) / (2^9 ⋅ 9^(-10))
Next, we apply the property a^(-n) = 1/(a^n) to convert the negative exponent to positive in the denominator:
= (2^(12) ⋅ 9^(-12)) / (2^9 ⋅ 1 / 9^(10))
Recall the property a/(b/c) = a*(c/b):
= (2^(12) ⋅ 9^(-12)) * (9^(10) / 2^9)
Now we multiply the numbers with the same base:
= 2^(12) ⋅ 9^(-12) * (9^(10) ⋅ 9^(-12))
Using the property a^n / a^m = a^(n-m), we have:
= 2^(12-9) ⋅ 9^(10-12)
= 2^3 ⋅ 9^(-2)
Lastly, we can rewrite 9^(-2) as 1/(9^2):
= 2^3 ⋅ 1/(9^2)
= 8 / 81
Therefore, an equivalent expression with only positive exponents is 8/81.
no its not. its not that.
I apologize for the mistake in my previous response. Let's simplify the expression correctly.
Given expression: (2^(-3) ⋅ 9^3)^(-4) / 2^9 ⋅ 9^(-10)
First, let's simplify the numerator: (2^(-3) ⋅ 9^3)^(-4)
Using the property (a^m)^n = a^(m*n), we can simplify this as follows:
= 2^((-3)*(-4)) ⋅ 9^(3*(-4))
= 2^(12) ⋅ 9^(-12)
Now, let's simplify the denominator: 2^9 ⋅ 9^(-10)
To convert the negative exponent in the denominator to positive, we can use the property a^(-n) = 1/(a^n):
= 2^9 ⋅ 1/(9^10)
The expression now becomes:
(2^(12) ⋅ 9^(-12)) / (2^9 ⋅ 1/(9^10))
Next, let's simplify further by multiplying the numbers with the same base:
= (2^(12-9)) ⋅ (9^(-12) ⋅ (9^10))
= 2^3 ⋅ 9^(-12+10)
= 2^3 ⋅ 9^(-2)
Finally, let's rewrite 9^(-2) as 1/(9^2):
= 2^3 ⋅ 1/(9^2)
= 8 / (9^2)
= 8 / 81
Therefore, the correct equivalent expression with only positive exponents is 8 / 81.