Solve for r 3646=1+5(4r+17)^(3/2)

$3646 = 1 + 5 {(4 r + 17)}^{\frac{3}{2}}$

Rewrite the equation as $1 + 5 {(4 r + 17)}^{\frac{3}{2}} = 3646$ .

$1 + 5 {(4 r + 17)}^{\frac{3}{2}} = 3646$

Move all terms not containing $r$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$5 {(4 r + 17)}^{\frac{3}{2}} = 3646 - 1$

Subtract $1$ from $3646$ .

$5 {(4 r + 17)}^{\frac{3}{2}} = 3645$

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(5 {(4 r + 17)}^{\frac{3}{2}})}^{\frac{2}{3}} = 3645^{\frac{2}{3}}$

Simplify the left side.

Simplify ${(5 {(4 r + 17)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the product rule to $5 {(4 r + 17)}^{\frac{3}{2}}$ .

$5^{\frac{2}{3}} {({(4 r + 17)}^{\frac{3}{2}})}^{\frac{2}{3}} = 3645^{\frac{2}{3}}$

Multiply the exponents in ${({(4 r + 17)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5^{\frac{2}{3}} {(4 r + 17)}^{\frac{3}{2} \cdot \frac{2}{3}} = 3645^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$5^{\frac{2}{3}} {(4 r + 17)}^{\frac{3}{2} \cdot \frac{2}{3}} = 3645^{\frac{2}{3}}$

Rewrite the expression.

$5^{\frac{2}{3}} {(4 r + 17)}^{\frac{1}{2} \cdot 2} = 3645^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5^{\frac{2}{3}} {(4 r + 17)}^{\frac{1}{2} \cdot 2} = 3645^{\frac{2}{3}}$

Rewrite the expression.

$5^{\frac{2}{3}} {(4 r + 17)}^{1} = 3645^{\frac{2}{3}}$

Simplify.

$5^{\frac{2}{3}} (4 r + 17) = 3645^{\frac{2}{3}}$

Apply the distributive property.

$5^{\frac{2}{3}} (4 r) + 5^{\frac{2}{3}} \cdot 17 = 3645^{\frac{2}{3}}$

Simplify the expression.

Move $17$ to the left of $5^{\frac{2}{3}}$ .

$5^{\frac{2}{3}} \cdot 4 r + 17 \cdot 5^{\frac{2}{3}} = 3645^{\frac{2}{3}}$

Move $4$ to the left of $5^{\frac{2}{3}}$ .

$4 \cdot 5^{\frac{2}{3}} r + 17 \cdot 5^{\frac{2}{3}} = 3645^{\frac{2}{3}}$

Reorder factors in $4 \cdot 5^{\frac{2}{3}} r + 17 \cdot 5^{\frac{2}{3}}$ .

$4 r \cdot 5^{\frac{2}{3}} + 17 \cdot 5^{\frac{2}{3}} = 3645^{\frac{2}{3}}$

Solve for $r$ .

Subtract $17 \cdot 5^{\frac{2}{3}}$ from both sides of the equation.

$4 r \cdot 5^{\frac{2}{3}} = 3645^{\frac{2}{3}} - 17 \cdot 5^{\frac{2}{3}}$

Divide each term in $(4 r \cdot 5^{\frac{2}{3}}) = 3645^{\frac{2}{3}} - 17 \cdot 5^{\frac{2}{3}}$ by $4 \cdot 5^{\frac{2}{3}}$ and simplify.

Divide each term in $(4 r \cdot 5^{\frac{2}{3}}) = 3645^{\frac{2}{3}} - 17 \cdot 5^{\frac{2}{3}}$ by $4 \cdot 5^{\frac{2}{3}}$ .

$\frac{(4 r \cdot 5^{\frac{2}{3}})}{4 \cdot 5^{\frac{2}{3}}} = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Simplify the left side.

Cancel the common factor.

$\frac{4 r \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{r \cdot 5^{\frac{2}{3}}}{5^{\frac{2}{3}}} = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{r \cdot 5^{\frac{2}{3}}}{5^{\frac{2}{3}}} = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Divide $r$ by $1$ .

$r = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Simplify the right side.

Simplify each term.

Cancel the common factor.

$r = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17 \cdot 5^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}}$

Rewrite the expression.

$r = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} + \frac{- 17}{4}$

Move the negative in front of the fraction.

$r = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} - \frac{17}{4}$

The result can be shown in multiple forms.

Exact Form:

$r = \frac{3645^{\frac{2}{3}}}{4 \cdot 5^{\frac{2}{3}}} - \frac{17}{4}$

Decimal Form:

$r = 16$

Solve for r 3646=1+5(4r+17)^(3/2)

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