Algebra Examples

$- 646 = - 3 {(65 - n)}^{\frac{3}{2}} + 2$

Step 1

Rewrite the equation as $- 3 {(65 - n)}^{\frac{3}{2}} + 2 = - 646$ .

$- 3 {(65 - n)}^{\frac{3}{2}} + 2 = - 646$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the equation.

$- 3 {(65 - n)}^{\frac{3}{2}} = - 646 - 2$

Step 2.2

Subtract $2$ from $- 646$ .

$- 3 {(65 - n)}^{\frac{3}{2}} = - 648$

Step 3

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

${(- 3 {(65 - n)}^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 4

Simplify the left side.

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Step 4.1

Simplify ${(- 3 {(65 - n)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 4.1.1

Apply the product rule to $- 3 {(65 - n)}^{\frac{3}{2}}$ .

${(- 3)}^{\frac{2}{3}} {({(65 - n)}^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.2

Multiply the exponents in ${({(65 - n)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 4.1.2.1

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

${(- 3)}^{\frac{2}{3}} {(65 - n)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

${(- 3)}^{\frac{2}{3}} {(65 - n)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.2.2.2

Rewrite the expression.

${(- 3)}^{\frac{2}{3}} {(65 - n)}^{\frac{1}{2} \cdot 2} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.2.3

Cancel the common factor of $2$ .

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Step 4.1.2.3.1

Cancel the common factor.

${(- 3)}^{\frac{2}{3}} {(65 - n)}^{\frac{1}{2} \cdot 2} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.2.3.2

Rewrite the expression.

${(- 3)}^{\frac{2}{3}} {(65 - n)}^{1} = {(- 648)}^{\frac{2}{3}}$

Step 4.1.3

Simplify.

${(- 3)}^{\frac{2}{3}} (65 - n) = {(- 648)}^{\frac{2}{3}}$

Step 4.1.4

Apply the distributive property.

${(- 3)}^{\frac{2}{3}} \cdot 65 + {(- 3)}^{\frac{2}{3}} (- n) = {(- 648)}^{\frac{2}{3}}$

Step 4.1.5

Reorder.

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Step 4.1.5.1

Move $65$ to the left of ${(- 3)}^{\frac{2}{3}}$ .

$65 \cdot {(- 3)}^{\frac{2}{3}} + {(- 3)}^{\frac{2}{3}} (- n) = {(- 648)}^{\frac{2}{3}}$

Step 4.1.5.2

Reorder factors in $65 {(- 3)}^{\frac{2}{3}} + {(- 3)}^{\frac{2}{3}} (- n)$ .

$65 {(- 3)}^{\frac{2}{3}} - n {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 5

Solve for $n$ .

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Step 5.1

Subtract $65 {(- 3)}^{\frac{2}{3}}$ from both sides of the equation.

$- n {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$

Step 5.2

Divide each term in $- n {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$ by $- {(- 3)}^{\frac{2}{3}}$ and simplify.

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Step 5.2.1

Divide each term in $- n {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$ by $- {(- 3)}^{\frac{2}{3}}$ .

$\frac{- n {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{n {(- 3)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{n {(- 3)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.2.3

Divide $n$ by $1$ .

$n = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$n = - \frac{{(- 648)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.2

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$n = - {(\frac{- 648}{- 3})}^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.3

Divide $- 648$ by $- 3$ .

$n = - 216^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.4

Rewrite $216$ as $6^{3}$ .

$n = - {(6^{3})}^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.5

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

$n = - 6^{3 (\frac{2}{3})} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.6

Cancel the common factor of $3$ .

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Step 5.2.3.1.6.1

Cancel the common factor.

$n = - 6^{3 (\frac{2}{3})} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.6.2

Rewrite the expression.

$n = - 6^{2} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.7

Raise $6$ to the power of $2$ .

$n = - 1 \cdot 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.8

Multiply $- 1$ by $36$ .

$n = - 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.9

Cancel the common factor.

$n = - 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Step 5.2.3.1.10

Rewrite the expression.

$n = - 36 + \frac{- 65}{- 1}$

Step 5.2.3.1.11

Move the negative one from the denominator of $\frac{- 65}{- 1}$ .

$n = - 36 - 1 \cdot - 65$

Step 5.2.3.1.12

Rewrite $- 1 \cdot - 65$ as $- - 65$ .

$n = - 36 - - 65$

Step 5.2.3.1.13

Multiply $- 1$ by $- 65$ .

$n = - 36 + 65$

Step 5.2.3.2

Add $- 36$ and $65$ .

$n = 29$