Which of the following is the equivalent expression to (2^-3 x 9^3)^-4/2^9 x 9^-10 with only positive exponents generated by applying the properties of integer exponents?
A 2^12x9^-12/2^9x9^-10
B 2^3/9^2
C 2^-7x9^-1/2^9x9^-10
D 2^3 x 9^2
To simplify the expression, let's apply the properties of exponentiation step by step:
(2^-3 x 9^3)^-4/2^9 x 9^-10
First, we can simplify the exponent of each base:
(2^-3 x (9^3)^1)^-4/2^9 x 9^-10
Now, we can apply the property (a^m)^n = a^(m*n):
(2^-3 x 9^(3*1))^-4/2^9 x 9^-10
Simplifying further:
(2^-3 x 9^3)^-4/2^9 x 9^-10
Now, let's simplify the expression inside the parentheses:
(8 x 9^3)^-4/2^9 x 9^-10
Next, we apply the property (a * b)^n = a^n * b^n:
8^-4 x (9^3)^-4/2^9 x 9^-10
Simplifying further:
8^-4 x 9^(-12)/2^9 x 9^-10
Now, let's apply the property a^-m = 1/a^m:
1/8^4 x 9^-12/2^9 x 9^-10
Next, apply the property 1/a^m = a^-m:
(8^4)^-1 x 9^-12/2^9 x 9^-10
(8^(-4))^-1 x 9^-12/2^9 x 9^-10
8^(4*-1) x 9^-12/2^9 x 9^-10
8^-4 x 9^-12/2^9 x 9^-10
Now, let's apply the property a^-m = 1/a^m again:
1/8^4 x 1/9^12/2^9 x 1/9^10
Next, let's apply the property (a * b) / (c * d) = a/c * b/d:
1/(8^4 x 9^12) x 2^(-9) x 1/9^10
Now, rewrite the expression using positive exponents:
2^-9 / (8^4 x 9^12 x 9^10)
Now, let's apply the property a^-m / a^n = a^(n-m):
2^(-9 - 4) x 9^(-12 - 10)
2^-13 x 9^-22
Finally, let's rewrite the expression using positive exponents:
1/2^13 x 1/9^22
So, the equivalent expression with only positive exponents is:
A) 2^12 x 9^-12 / 2^9 x 9^-10
maybe (you need to be more careful with parentheses when asking this sort of question)
(9^3/2^3)^-4 * 9^10 / 2^9
9^-12/2^-12 * 9^10 / 2^9
9^-2 / 2^-3
2^3 / 9^2
(which is B)
by the way = 8 / 81
You are correct, my previous response overlooked the proper placement of parentheses. The correct simplification is indeed:
(9^3/2^3)^-4 * 9^10 / 2^9
To simplify, we can apply the properties of exponents:
(9^-12/2^-12) * 9^10 / 2^9
Using the property a^-m = 1/a^m:
(2^12/9^12) * 9^10 / 2^9
Simplifying further:
2^12 * (9^10 * 9^-12) / (2^9 * 9^12)
Applying the property a^m * a^n = a^(m+n):
2^12 * 9^(10+(-12)) / (2^9 * 9^12)
Simplifying exponents:
2^12 * 9^-2 / (2^9 * 9^12)
Using the property a^-m = 1/a^m again:
2^12 / (2^9 * 9^2)
Applying the property a^m / a^n = a^(m-n):
2^(12-9) / 9^2
Simplifying further:
2^3 / 9^2
Therefore, the equivalent expression with only positive exponents is:
B) 2^3 / 9^2, which simplifies to 8/81.