Algebra Examples

$- 5126 = - 6 - 5 {(3 x + 22)}^{\frac{5}{3}}$

Step 1

Rewrite the equation as $- 6 - 5 {(3 x + 22)}^{\frac{5}{3}} = - 5126$ .

$- 6 - 5 {(3 x + 22)}^{\frac{5}{3}} = - 5126$

Step 2

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

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Step 2.1

Add $6$ to both sides of the equation.

$- 5 {(3 x + 22)}^{\frac{5}{3}} = - 5126 + 6$

Step 2.2

Add $- 5126$ and $6$ .

$- 5 {(3 x + 22)}^{\frac{5}{3}} = - 5120$

Step 3

Raise each side of the equation to the power of $\frac{3}{5}$ to eliminate the fractional exponent on the left side.

${(- 5 {(3 x + 22)}^{\frac{5}{3}})}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4

Simplify the left side.

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Step 4.1

Simplify ${(- 5 {(3 x + 22)}^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 4.1.1

Apply the product rule to $- 5 {(3 x + 22)}^{\frac{5}{3}}$ .

${(- 5)}^{\frac{3}{5}} {({(3 x + 22)}^{\frac{5}{3}})}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.2

Multiply the exponents in ${({(3 x + 22)}^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 4.1.2.1

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

${(- 5)}^{\frac{3}{5}} {(3 x + 22)}^{\frac{5}{3} \cdot \frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.2.2

Cancel the common factor of $5$ .

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Step 4.1.2.2.1

Cancel the common factor.

${(- 5)}^{\frac{3}{5}} {(3 x + 22)}^{\frac{5}{3} \cdot \frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.2.2.2

Rewrite the expression.

${(- 5)}^{\frac{3}{5}} {(3 x + 22)}^{\frac{1}{3} \cdot 3} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.2.3

Cancel the common factor of $3$ .

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Step 4.1.2.3.1

Cancel the common factor.

${(- 5)}^{\frac{3}{5}} {(3 x + 22)}^{\frac{1}{3} \cdot 3} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.2.3.2

Rewrite the expression.

${(- 5)}^{\frac{3}{5}} {(3 x + 22)}^{1} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.3

Simplify.

${(- 5)}^{\frac{3}{5}} (3 x + 22) = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.4

Simplify by multiplying through.

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Step 4.1.4.1

Apply the distributive property.

${(- 5)}^{\frac{3}{5}} (3 x) + {(- 5)}^{\frac{3}{5}} \cdot 22 = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.4.2

Move $22$ to the left of ${(- 5)}^{\frac{3}{5}}$ .

${(- 5)}^{\frac{3}{5}} \cdot 3 x + 22 \cdot {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.5

Move $3$ to the left of ${(- 5)}^{\frac{3}{5}}$ .

$3 {(- 5)}^{\frac{3}{5}} x + 22 {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 4.1.6

Reorder factors in $3 {(- 5)}^{\frac{3}{5}} x + 22 {(- 5)}^{\frac{3}{5}}$ .

$3 x {(- 5)}^{\frac{3}{5}} + 22 {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}}$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $22 {(- 5)}^{\frac{3}{5}}$ from both sides of the equation.

$3 x {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}} - 22 {(- 5)}^{\frac{3}{5}}$

Step 5.2

Divide each term in $3 x {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}} - 22 {(- 5)}^{\frac{3}{5}}$ by $3 {(- 5)}^{\frac{3}{5}}$ and simplify.

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Step 5.2.1

Divide each term in $3 x {(- 5)}^{\frac{3}{5}} = {(- 5120)}^{\frac{3}{5}} - 22 {(- 5)}^{\frac{3}{5}}$ by $3 {(- 5)}^{\frac{3}{5}}$ .

$\frac{3 x {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor.

$\frac{3 x {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

Step 5.2.2.2

Rewrite the expression.

$\frac{x {(- 5)}^{\frac{3}{5}}}{{(- 5)}^{\frac{3}{5}}} = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

Step 5.2.2.3

Cancel the common factor.

$\frac{x {(- 5)}^{\frac{3}{5}}}{{(- 5)}^{\frac{3}{5}}} = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

Step 5.2.2.4

Divide $x$ by $1$ .

$x = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

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

Cancel the common factor.

$x = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22 {(- 5)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}}$

Step 5.2.3.1.2

Rewrite the expression.

$x = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} + \frac{- 22}{3}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

$x = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} - \frac{22}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{{(- 5120)}^{\frac{3}{5}}}{3 {(- 5)}^{\frac{3}{5}}} - \frac{22}{3}$

Decimal Form:

$x = 14$