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Simplify. Assume all variables are positive. $\newline$ $\frac{t^{\frac{3}{2}}}{t^{\frac{3}{2}} \cdot t^{\frac{1}{2}}}$ $\newline$ 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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Simplify radical expressions with variables II

Full solution

Q. Simplify. Assume all variables are positive.

\newline

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

______

Apply Exponent Properties: Write down the expression and apply the properties of exponents. $\newline$ We have the expression $t^{3/2}/(t^{3/2} \cdot t^{1/2})$ . According to the properties of exponents, when we multiply terms with the same base, we add their exponents. So, $t^{3/2} \cdot t^{1/2}$ becomes $t^{3/2 + 1/2}$ .
Add Exponents of t: Add the exponents of t. $\newline$ $t^{\frac{3}{2} + \frac{1}{2}} = t^{\frac{2}{2}} = t^1 = t$ . $\newline$ So, the expression in the denominator simplifies to $t$ .
Divide Terms with Same Base: Divide the terms with the same base. $\newline$ Now we have $t^{\frac{3}{2}} / t$ . Since $t = t^1$ , we can write this as $t^{\frac{3}{2}} / t^1$ . According to the properties of exponents, when we divide terms with the same base, we subtract their exponents. So, $t^{\frac{3}{2}} / t^1$ becomes $t^{\frac{3}{2} - 1}$ .
Subtract Exponents of t: Subtract the exponents of t. $\newline$ $t^{\frac{3}{2} - 1} = t^{\frac{3}{2} - \frac{2}{2}} = t^{\frac{1}{2}}$ . $\newline$ So, the expression simplifies to $t^{\frac{1}{2}}$ .
Write Final Answer: Write the final answer. $\newline$ Since $t^{1/2}$ is already in its simplest form and all exponents are positive, this is our final answer.

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