how do i simplify the following expression ,writing the answer with a positive exponent only?
(yz^-2)^4)(2y^-3z^-4)3/4y^9z^-5
(YZ^-2)^4(2Y^-3Z^-4)^3 / (4Y^9Z^-5 =
(Y^4Z^-8)(8Y^-9Z^-12) / (4Y^9Z^-5)=
8Y^-5Z^-20 / 4Y^9Z^-5 =
2Y^-14Z^-15 = 2 / Y^14Z^15
To simplify the given expression and write the answer with a positive exponent only, we need to apply the laws of exponents. Here's the step-by-step process:
Step 1: Simplify the expression within the first parentheses.
Inside the first parentheses, we have (yz^(-2))^4. To simplify this, we use the power of a power rule: raise the base (yz^(-2)) to the power of the exponent (4), which results in y^4z^(-8).
Step 2: Simplify the expression within the second parentheses.
Inside the second parentheses, we have 2y^(-3)z^(-4). To simplify this, we leave the numerical coefficient (2) as it is and apply the product of powers rule separately to y and z. For y, we multiply exponents: y^(-3) becomes y^(-3 * 1) = y^(-3). Similarly, for z, z^(-4) remains z^(-4).
Step 3: Simplify the remaining part of the expression.
The remaining part of the expression is 3/4y^9z^(-5). At this point, we multiply the coefficients (3 and 4) to get 3/4. Then, using the product of powers rule, we multiply the variables with the same bases (y and z). For y, y^9 remains y^9. For z^(-5), we need to multiply exponents: z^(-5) becomes z^(-5 * 1) = z^(-5).
Step 4: Combine the simplified parts.
Now that we have simplified all the parts, let's combine them:
(yz^(-2))^4 = y^4z^(-8)
2y^(-3)z^(-4) = 2y^(-3)z^(-4)
3/4y^9z^(-5) = (3/4)y^9z^(-5)
Step 5: Multiply all the combined parts.
To get the final answer, we multiply all the combined parts together:
(y^4z^(-8)) * (2y^(-3)z^(-4)) * ((3/4)y^9z^(-5))
Multiplying the coefficients, we get (3/4) * 2 = 6/4 = 3/2.
Using the product of powers rule for the variables, we add the exponents for each variable:
y^(4 + (-3) + 9) = y^10
z^((-8) + (-4) + (-5)) = z^(-17)
Therefore, the final simplified expression, with a positive exponent only, is:
(3/2)y^10z^(-17)