Simplify. Assume all variables are positive. $\newline$ $\frac{b^{\frac{7}{4}}}{b^{\frac{3}{4}} \cdot b^{\frac{7}{4}}}$ $\newline$ 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. $\newline$ ______

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Precalculus

Simplify expressions involving rational exponents

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Q. Simplify. Assume all variables are positive.

\newline

\frac{b^{\frac{7}{4}}}{b^{\frac{3}{4}} \cdot b^{\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 Given Expression: Identify the given expression and the properties of exponents that will be used to simplify it. $\newline$ The expression is $b^{\frac{7}{4}} / (b^{\frac{3}{4}} * b^{\frac{7}{4}})$ . We will use the properties of exponents, specifically the quotient rule $(a^m / a^n = a^{m-n})$ and the product rule $(a^m * a^n = a^{m+n})$ .
Apply Product Rule: Apply the product rule to the denominator of the expression. $\newline$ According to the product rule, when multiplying two exponents with the same base, you add the exponents: $b^{3/4} \times b^{7/4} = b^{3/4 + 7/4}$ .
Calculate Sum of Exponents: Calculate the sum of the exponents in the denominator. $\newline$ Adding the exponents: $\frac{3}{4} + \frac{7}{4} = \frac{10}{4}$ , which simplifies to $\frac{5}{2}$ . So, $b^{\frac{3}{4}} \times b^{\frac{7}{4}} = b^{\frac{5}{2}}$ .
Rewrite with Simplified Denominator: Rewrite the original expression with the simplified denominator. $\newline$ The expression now is $b^{\frac{7}{4}} / b^{\frac{5}{2}}$ .
Apply Quotient Rule: Apply the quotient rule to simplify the expression. $\newline$ According to the quotient rule, when dividing two exponents with the same base, you subtract the exponents: $b^{\frac{7}{4}} / b^{\frac{5}{2}} = b^{\frac{7}{4} - \frac{5}{2}}$ .
Convert Exponents: Convert the exponents to have a common denominator before subtracting. $\newline$ The common denominator for $4$ and $2$ is $4$ . Convert $\frac{5}{2}$ to $\frac{10}{4}$ so that both exponents have the same denominator: $b^{\frac{7}{4} - \frac{10}{4}}$ .
Calculate Difference of Exponents: Calculate the difference of the exponents. $\newline$ Subtracting the exponents: $\frac{7}{4} - \frac{10}{4} = -\frac{3}{4}$ . So, $b^{\frac{7}{4}} / b^{\frac{5}{2}} = b^{-\frac{3}{4}}$ .
Rewrite as Reciprocal: Since negative exponents indicate the reciprocal, rewrite $b^{-3/4}$ as $1/b^{3/4}$ . The expression $b^{-3/4}$ is equivalent to $1/b^{3/4}$ according to the negative exponent rule.