Question

7 - 2 division properties of exponents
simplify each expression. assume that no denominator equals zero.

1. \\(\frac{(3x)^{0}}{2a}\\)
2. \\((\frac{3xy^{3}}{2z})^{3}\\)
3. \\(\frac{12y^{-4}}{3y^{-5}}\\)
4. \\(a^{-3}b^{0}c^{6}\\)
5. \\(\frac{-15x^{7}y^{8}z^{4}}{-45x^{3}y^{5}z^{3}}\\)
6. \\(\frac{(3x^{-1})^{-2}}{(3x^{2})^{-2}}\\)
7. \\((\frac{6xy^{11}z^{9}}{48x^{6}yz^{-7}})^{0}\\)
8. \\((\frac{12}{2})(\frac{x}{y^{5}})(\frac{y^{4}}{x})\\)
9. geometry the area of a rectangle is 25x^{2}y^{4} square feet. the width of the rectangle is 5xy feet. what is the length of the rectangle?

Explanation:

Step1: Recall area formula

The area formula of a rectangle is $A = lw$, where $A$ is area, $l$ is length and $w$ is width. We need to find $l$, so $l=\frac{A}{w}$.

Step2: Substitute given values

Given $A = 25x^{2}y^{4}$ and $w = 5xy$. Then $l=\frac{25x^{2}y^{4}}{5xy}$.

Step3: Use division - property of exponents

For the coefficient, $\frac{25}{5}=5$. For the $x$ - terms, using the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$, we have $\frac{x^{2}}{x}=x^{2-1}=x$. For the $y$ - terms, $\frac{y^{4}}{y}=y^{4 - 1}=y^{3}$.

Step4: Combine results

$l = 5xy^{3}$.

Answer:

$5xy^{3}$