Solve for S S(t)=2t^(3/2)-9t+6

$S (t) = 2 t^{\frac{3}{2}} - 9 t + 6$

Step 1

Divide each term in $S t = 2 t^{\frac{3}{2}} - 9 t + 6$ by $t$ .

$\frac{S t}{t} = \frac{2 t^{\frac{3}{2}}}{t} + \frac{- 9 t}{t} + \frac{6}{t}$

Step 2

Simplify the left side.

Cancel the common factor of $t$ .

Cancel the common factor.

$\frac{S t}{t} = \frac{2 t^{\frac{3}{2}}}{t} + \frac{- 9 t}{t} + \frac{6}{t}$

Divide $S$ by $1$ .

$S = \frac{2 t^{\frac{3}{2}}}{t} + \frac{- 9 t}{t} + \frac{6}{t}$

Step 3

Simplify the right side.

Simplify each term.

Move $t^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$S = 2 t^{\frac{3}{2}} t^{- 1} + \frac{- 9 t}{t} + \frac{6}{t}$

Multiply $t^{\frac{3}{2}}$ by $t^{- 1}$ by adding the exponents.

Move $t^{- 1}$ .

$S = 2 (t^{- 1} t^{\frac{3}{2}}) + \frac{- 9 t}{t} + \frac{6}{t}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$S = 2 t^{- 1 + \frac{3}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$S = 2 t^{- 1 \cdot \frac{2}{2} + \frac{3}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Combine $- 1$ and $\frac{2}{2}$ .

$S = 2 t^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Combine the numerators over the common denominator.

$S = 2 t^{\frac{- 1 \cdot 2 + 3}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$S = 2 t^{\frac{- 2 + 3}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Add $- 2$ and $3$ .

$S = 2 t^{\frac{1}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Cancel the common factor of $t$ .

Cancel the common factor.

$S = 2 t^{\frac{1}{2}} + \frac{- 9 t}{t} + \frac{6}{t}$

Divide $- 9$ by $1$ .

$S = 2 t^{\frac{1}{2}} - 9 + \frac{6}{t}$