Simplify. Assume all variables are positive. $\newline$ $\frac{u^{\frac{9}{4}}}{u^{\frac{11}{4}} \cdot u^{\frac{9}{4}}}$ $\newline$ Write your answer in the form $A$ or $\frac{A}{B}$ , 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{9}{4}}}{u^{\frac{11}{4}} \cdot u^{\frac{9}{4}}}

\newline

Write your answer in the form

A

\frac{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 Expression: Identify the expression to be simplified. $\newline$ We have the expression $u^{\frac{9}{4}} / (u^{\frac{11}{4}} * u^{\frac{9}{4}})$ . We need to simplify this expression by using the properties of exponents.
Combine Exponents: Combine the exponents in the denominator using the product rule for exponents. $\newline$ The product rule states that $a^m \times a^n = a^{m+n}$ when the bases are the same. So, we combine the exponents in the denominator: $\newline$ $u^{11/4} \times u^{9/4} = u^{11/4 + 9/4} = u^{20/4} = u^5$ .
Rewrite Expression: Rewrite the original expression with the simplified denominator. $\newline$ Now that we have simplified the denominator, we rewrite the expression as: $\newline$ $u^{\frac{9}{4}} / u^{5}$ .
Simplify Using Quotient Rule: Simplify the expression using the quotient rule for exponents. $\newline$ The quotient rule states that $a^m / a^n = a^{(m-n)}$ when the bases are the same. So, we subtract the exponents: $\newline$ $u^{9/4} / u^5 = u^{9/4 - 20/4} = u^{-11/4}$ .
Make Exponent Positive: Since we want all exponents to be positive, we rewrite the expression with a positive exponent. $\newline$ To make the exponent positive, we take the reciprocal of the base: $\newline$ $u^{(-11/4)} = 1/u^{(11/4)}$ .