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

Full solution

Q. Simplify. Assume all variables are positive.

\newline

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

______

Identify Expression: Identify the expression to simplify. $\newline$ We have the expression $t^{\frac{3}{2}} / (t^{\frac{5}{2}} * t^{\frac{3}{2}})$ .
Combine Exponents: Apply the property of exponents to combine the exponents in the denominator. $\newline$ When multiplying with the same base, we add the exponents: $t^{a} \times t^{b} = t^{a+b}$ . $\newline$ So, $t^{\frac{5}{2}} \times t^{\frac{3}{2}} = t^{\frac{5}{2} + \frac{3}{2}} = t^{\frac{8}{2}} = t^4$ .
Rewrite with Simplified Denominator: Rewrite the expression with the simplified denominator. $\newline$ Now we have $t^{\frac{3}{2}} / t^4$ .
Apply Exponent Property for Division: Apply the property of exponents for division with the same base. $\newline$ When dividing with the same base, we subtract the exponents: $t^{a} / t^{b} = t^{a-b}$ . $\newline$ So, $t^{3/2} / t^{4} = t^{3/2 - 4} = t^{3/2 - 8/2} = t^{-5/2}.$
Rewrite Negative Exponent: Since we want the exponent to be positive, we can rewrite $t^{(-5/2)}$ as $1 / t^{(5/2)}$ . $\newline$ This is because $t^{(-a)} = 1 / t^{(a)}$ for any positive exponent $a$ .