Precalculus Examples

$\sqrt[3]{10 x^{2} + 5 x} = \sqrt[3]{7 x^{2} + 2}$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{10 x^{2} + 5 x}}^{3} = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10 x^{2} + 5 x}$ as ${(10 x^{2} + 5 x)}^{\frac{1}{3}}$ .

${({(10 x^{2} + 5 x)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(10 x^{2} + 5 x)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(10 x^{2} + 5 x)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(10 x^{2} + 5 x)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(10 x^{2} + 5 x)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(10 x^{2} + 5 x)}^{1} = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2.2.1.2

Simplify.

$10 x^{2} + 5 x = {\sqrt[3]{7 x^{2} + 2}}^{3}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Rewrite ${\sqrt[3]{7 x^{2} + 2}}^{3}$ as $7 x^{2} + 2$ .

Tap for more steps...

Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 x^{2} + 2}$ as ${(7 x^{2} + 2)}^{\frac{1}{3}}$ .

$10 x^{2} + 5 x = {({(7 x^{2} + 2)}^{\frac{1}{3}})}^{3}$

Step 2.3.1.2

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

$10 x^{2} + 5 x = {(7 x^{2} + 2)}^{\frac{1}{3} \cdot 3}$

Step 2.3.1.3

Combine $\frac{1}{3}$ and $3$ .

$10 x^{2} + 5 x = {(7 x^{2} + 2)}^{\frac{3}{3}}$

Step 2.3.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.1.4.1

Cancel the common factor.

$10 x^{2} + 5 x = {(7 x^{2} + 2)}^{\frac{3}{3}}$

Step 2.3.1.4.2

Rewrite the expression.

$10 x^{2} + 5 x = {(7 x^{2} + 2)}^{1}$

Step 2.3.1.5

Simplify.

$10 x^{2} + 5 x = 7 x^{2} + 2$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.1.1

Subtract $7 x^{2}$ from both sides of the equation.

$10 x^{2} + 5 x - 7 x^{2} = 2$

Step 3.1.2

Subtract $7 x^{2}$ from $10 x^{2}$ .

$3 x^{2} + 5 x = 2$

Step 3.2

Subtract $2$ from both sides of the equation.

$3 x^{2} + 5 x - 2 = 0$

Step 3.3

Factor by grouping.

Tap for more steps...

Step 3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 5$ .

Tap for more steps...

Step 3.3.1.1

Factor $5$ out of $5 x$ .

$3 x^{2} + 5 (x) - 2 = 0$

Step 3.3.1.2

Rewrite $5$ as $- 1$ plus $6$

$3 x^{2} + (- 1 + 6) x - 2 = 0$

Step 3.3.1.3

Apply the distributive property.

$3 x^{2} - 1 x + 6 x - 2 = 0$

Step 3.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.3.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 1 x) + 6 x - 2 = 0$

Step 3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 1) + 2 (3 x - 1) = 0$

Step 3.3.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x + 2) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x - 1 = 0$

$x + 2 = 0$

Step 3.5

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.5.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.5.2

Solve $3 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 3.5.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.5.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

Tap for more steps...

Step 3.5.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 3.6

Set $x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.7

The final solution is all the values that make $(3 x - 1) (x + 2) = 0$ true.

$x = \frac{1}{3}, - 2$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, - 2$

Decimal Form:

$x = 0.\overline{3}, - 2$