Simplify. Assume all variables are positive. $\newline$ $\frac{d^{\frac{5}{4}}}{d^{\frac{5}{4}} \cdot d^{\frac{7}{4}}}$ $\newline$ 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{d^{\frac{5}{4}}}{d^{\frac{5}{4}} \cdot d^{\frac{7}{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 simplify. $\newline$ We have the expression $d^{\frac{5}{4}} / (d^{\frac{5}{4}} * d^{\frac{7}{4}})$ . We need to simplify this expression by combining the exponents in the denominator and then dividing the numerator by the simplified denominator.
Combine Exponents: Combine the exponents in the denominator using the property of exponents that states $a^{m} \times a^{n} = a^{m+n}$ . $d^{\frac{5}{4}} \times d^{\frac{7}{4}} = d^{\frac{5}{4} + \frac{7}{4}} = d^{\frac{12}{4}} = d^3$ .
Rewrite Expression: Rewrite the original expression with the simplified denominator. The expression now becomes $d^{\frac{5}{4}} / d^3$ .
Simplify Exponents: Simplify the expression by subtracting the exponents in the denominator from the exponent in the numerator using the property $a^{m} / a^{n} = a^{m-n}$ . $d^{5/4} / d^3 = d^{5/4 - 12/4} = d^{-7/4}$ .
Rewrite with Positive Exponent: Since we want all exponents to be positive, we can rewrite the expression with a positive exponent by using the property $a^{-n} = \frac{1}{a^n}$ . $\newline$ $d^{-\frac{7}{4}} = \frac{1}{d^{\frac{7}{4}}}$ .