Simplify the following expression. Write the result using positive exponents.
(-2xy^-3)^-5 / (xy^-1)^-1
Using your exponents rules:
(-2)^5 x^5 y^15 * (xy^-1)^1
= -32 x^6 y^14
Hmmm. You seem to have (almost) lost a minus sign
(-2xy^-3)^-5 / (xy^-1)^-1
= -1/32 x^-4 y^14
go with Steve's answer, yes I missed a minus sign
To simplify the given expression, let's start by working on the numerators and the denominators separately.
Let's begin by simplifying the first exponent in the numerator: (-2xy^(-3))^(-5). To do this, we apply the power of a power rule, which states that (a^b)^c = a^(b * c). In this case, we will multiply the exponents:
(-2xy^(-3))^(-5) = -2^(-5) * (x^1)^(-5) * (y^(-3))^(-5).
Now, simplify each term individually:
-2^(-5) = -1/(2^5) = -1/32.
(x^1)^(-5) = x^(-5) = 1/x^5.
(y^(-3))^(-5) = (y^(-3 * -5)) = y^15.
Next, let's simplify the exponent in the denominator: (xy^(-1))^(-1). We can also use the power of a power rule here:
(xy^(-1))^(-1) = x^(-1) * (y^(-1))^(-1).
Simplify each term:
x^(-1) = 1/x,
(y^(-1))^(-1) = y^(1) = y.
Now, let's substitute the simplified expressions back into the original expression:
(-2xy^-3)^-5 / (xy^-1)^-1 = (-1/32)(1/x^5)(y^15) / (1/x)(y).
To divide fractions, we multiply by the reciprocal of the denominator:
= (-1/32)(1/x^5)(y^15)(x) / (1)(y).
Cancel out the common factors:
= (-1/32)(1/x^4)(y^14).
Finally, we can rewrite the expression using positive exponents:
= -1/(32x^4y^14).
So, the simplified expression, with positive exponents, is -1/(32x^4y^14).