Simplify. Assume all variables are positive. $\newline$ $\frac{b^{\frac{3}{2}}}{b^{\frac{3}{2}} \cdot b^{\frac{5}{2}}}$ $\newline$ 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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Algebra 2

Simplify radical expressions with variables II

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Q. Simplify. Assume all variables are positive.

\newline

\frac{b^{\frac{3}{2}}}{b^{\frac{3}{2}} \cdot b^{\frac{5}{2}}}

\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

______

Write Quotient Rule Application: Write down the expression and apply the quotient rule for exponents. $\newline$ The quotient rule states that when dividing like bases, you subtract the exponents. $\newline$ Expression: $b^{\frac{3}{2}} / (b^{\frac{3}{2}} \cdot b^{\frac{5}{2}})$
Simplify Denominator with Product Rule: Simplify the denominator using the product rule for exponents. The product rule states that when multiplying like bases, you add the exponents. Denominator: $b^{\frac{3}{2}} \times b^{\frac{5}{2}} = b^{\frac{3}{2} + \frac{5}{2}} = b^{\frac{8}{2}} = b^4$
Rewrite Expression with Simplified Denominator: Rewrite the expression with the simplified denominator. $\newline$ Expression: $\frac{b^{\frac{3}{2}}}{b^4}$
Apply Quotient Rule to Expression: Apply the quotient rule for exponents to the expression. $\newline$ Subtract the exponents: $(\frac{3}{2}) - 4 = (\frac{3}{2}) - (\frac{8}{2}) = -\frac{5}{2}$ $\newline$ Expression: $b^{-\frac{5}{2}}$
Rewrite Expression with Positive Exponent: Since we want positive exponents, rewrite the expression with a positive exponent. $\newline$ To make the exponent positive, we can take the reciprocal of the base. $\newline$ Expression: $\frac{1}{b^{\frac{5}{2}}}$
Check Final Expression: Check the final expression to ensure it meets the requirements. $\newline$ The final expression is in the form $\frac{A}{B}$ , where $A$ is $1$ and $B$ is $b^{\frac{5}{2}}$ , and there are no variables in common in the numerator and denominator. All exponents are positive.