Simplify each expression so that no negative exponents appear in the final result.
Assume that all variables represent nonzero real numbers.
1. (12k^-2(k^-3)^-4)/(6k^5)
2. (3rs^-2)/(3^2r^2s^-4)
(12k^-2 (k^-3)^-4)/(6k^5)
= (12k^-2 * k^12)/(6k^5)
= (12k^10)/(6k^5)
= 12/6 * k^10/k^5
= 2k^5
What do you get for the other one?
To simplify these expressions and get rid of negative exponents, we can use the properties of exponents.
1. (12k^(-2)((k^(-3))^(-4))/(6k^5)
First, let's simplify the innermost parentheses, (k^(-3))^(-4). To do this, we need to apply the power of a power rule, which states that (a^b)^c is equal to a^(b * c).
So, (k^(-3))^(-4) is equal to k^((-3) * (-4)) = k^12.
Now the expression becomes: (12k^(-2)(k^12))/(6k^5)
Next, we can simplify the expression by dividing the coefficients and subtracting the exponents if they have the same base.
(12/6)(k^(-2+12))/(k^5) = 2k^10/k^5
Now, we can subtract the exponents by dividing: 2k^(10-5) = 2k^5
Therefore, the simplified expression is 2k^5.
2. (3rs^(-2))/(3^2r^2s^(-4))
First, let's simplify the expression in the numerator. We can rewrite s^(-2) as 1/s^2.
So, (3r(1/s^2))/(3^2r^2s^(-4))
Next, let's simplify the expression in the denominator. We can rewrite s^(-4) as 1/s^4.
Now, the expression becomes: (3r(1/s^2))/(9r^2(1/s^4))
To simplify further, we can multiply the numerators and multiply the denominators.
(3r/s^2) / (9r^2/s^4)
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction.
(3r/s^2) * (s^4/9r^2)
Now, let's simplify by multiplying the numerators and the denominators.
(3r * s^4)/(s^2 * 9r^2)
Finally, we can cancel out common factors and simplify further.
3r * (s^4/s^2)/(9r^2)
Using the rule that s^(a-b) is equal to s^a/s^b, we can simplify s^4/s^2 as s^(4-2) = s^2.
Therefore, the simplified expression is:
(3r * s^2)/(9r^2)
And since 3 and 9 have a common factor of 3, we can simplify further:
r * (s^2)/(3r^2)
The final simplified expression is:
s^2/(3r)