Question

2. (12 points) Simplify the following expressions. The final answer should not involve a fraction. For

example, x/y is not in final form, but xy-¹ is.

Answer 1

`\large\text{1)\;\;$(x^(-3)y^4z^7)/(x^3y^(-3)z^5)=\boxed{x^(-6)y^7z^2}$}`

`\large\text{2)\;\;$((uv^3)^2)/((u^3v)^2)=\boxed{u^(-4)v^4}$}`

Explanation:

To simplify the given rational expressions, use the following exponent rules:

`\large\boxed{\begin{array}{l}\underline{\rm Exponent\;rules}\\\\(1)/(a^n)=a^(-n)\\\\a^b \cdot a^c=a^(b+c)\\\\(a^b)^c=a^(bc)\\\\(a^bc^d)^n=a^(bn)c^(dn)\end{array}}`

Question 1

`\large\begin{aligned}(x^(-3)y^4z^7)/(x^3y^(-3)z^5)&= x^(-3)x^(-3)y^4y^(-(-3))z^7z^{-5\\\\&= x^(-3)x^(-3)y^4y^(3)z^7z^{-5\\\\&=x^((-3+(-3)))y^((4+3))z^((7+(-5)))\\\\&=x^((-3-3))y^((4+3))z^((7-5))\\\\&=x^(-6)y^7z^2\end{aligned}`

Question 2

`\large\begin{aligned}((uv^3)^2)/((u^3v)^2)&=((u)^2 \cdot (v^3)^2)/((u^3)^2 \cdot (v)^2)\\\\&=(u^2v^6)/(u^6v^2)\\\\&=u^2u^(-6)v^6v^(-2)\\\\&=u^((2+(-6)))v^((6+(-2)))\\\\&=u^((2-6))v^((6-2))\\\\&=u^(-4)v^4\end{aligned}`