Simplify. Assume all variables are positive. $\newline$ $\frac{u^{\frac{3}{2}}}{u^{\frac{3}{2}} \cdot u^{\frac{1}{2}}}$ $\newline$ Write your answer in the form $A$ or $A/B$ , where $A$ and $B$ are constants or variable expressions that have no variables in common. All exponents in your answer should be positive. $\newline$ ______

Full solution

Q. Simplify. Assume all variables are positive.

\newline

\frac{u^{\frac{3}{2}}}{u^{\frac{3}{2}} \cdot u^{\frac{1}{2}}}

\newline

Write your answer in the form

A

A/B

, where

A

and

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

______

Identify and apply quotient rule: Identify the expression and apply the quotient rule for exponents. The expression is $u^{\frac{3}{2}}/(u^{\frac{3}{2}} \cdot u^{\frac{1}{2}})$ . According to the quotient rule, when we divide like bases, we subtract the exponents: $a^m / a^n = a^{m-n}$ .
Apply product rule to denominator: Apply the product rule for exponents to the denominator. $\newline$ The product rule states that when we multiply like bases, we add the exponents: $a^m \times a^n = a^{(m+n)}$ . So, $u^{\frac{3}{2}} \times u^{\frac{1}{2}} = u^{(\frac{3}{2} + \frac{1}{2})}$ .
Perform exponent addition: Perform the addition in the exponent. $\newline$ Add the exponents in the denominator: $\frac{3}{2} + \frac{1}{2} = \frac{4}{2} = 2$ . So, $u^{\frac{3}{2}} \times u^{\frac{1}{2}} = u^2$ .
Apply quotient rule to original expression: Apply the quotient rule to the original expression. $\newline$ Now we have $u^{\frac{3}{2}} / u^2$ . Using the quotient rule: $u^{\frac{3}{2} - 2} = u^{\frac{3}{2} - \frac{4}{2}}$ .
Perform exponent subtraction: Perform the subtraction in the exponent. $\newline$ Subtract the exponents: $\frac{3}{2} - \frac{4}{2} = -\frac{1}{2}$ . So, $u^{\frac{3}{2}} / u^{2} = u^{-\frac{1}{2}}$ .
Write final answer: Write the final answer with a positive exponent. $\newline$ Since we want the exponent to be positive, we can write $u^{-1/2}$ as $1/u^{1/2}$ .