Basic Math Examples

$\sqrt{7 w} - 2 = \sqrt{4 w} + 10$

Step 1

Solve for $\sqrt{7 w}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

$\sqrt{7 w} - 2 = \sqrt{2^{2} w} + 10$

Step 1.1.2

Pull terms out from under the radical.

$\sqrt{7 w} - 2 = 2 \sqrt{w} + 10$

Step 1.2

Move all terms not containing $\sqrt{7 w}$ to the right side of the equation.

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Step 1.2.1

Add $2$ to both sides of the equation.

$\sqrt{7 w} = 2 \sqrt{w} + 10 + 2$

Step 1.2.2

Add $10$ and $2$ .

$\sqrt{7 w} = 2 \sqrt{w} + 12$

Step 2

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

${\sqrt{7 w}}^{2} = {(2 \sqrt{w} + 12)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({(7 w)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{w} + 12)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(7 w)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(7 w)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(7 w)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{w} + 12)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(7 w)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{w} + 12)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(7 w)}^{1} = {(2 \sqrt{w} + 12)}^{2}$

Step 3.2.1.2

Simplify.

$7 w = {(2 \sqrt{w} + 12)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(2 \sqrt{w} + 12)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(2 \sqrt{w} + 12)}^{2}$ as $(2 \sqrt{w} + 12) (2 \sqrt{w} + 12)$ .

$7 w = (2 \sqrt{w} + 12) (2 \sqrt{w} + 12)$

Step 3.3.1.2

Expand $(2 \sqrt{w} + 12) (2 \sqrt{w} + 12)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$7 w = 2 \sqrt{w} (2 \sqrt{w} + 12) + 12 (2 \sqrt{w} + 12)$

Step 3.3.1.2.2

Apply the distributive property.

$7 w = 2 \sqrt{w} (2 \sqrt{w}) + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w} + 12)$

Step 3.3.1.2.3

Apply the distributive property.

$7 w = 2 \sqrt{w} (2 \sqrt{w}) + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $2 \sqrt{w} (2 \sqrt{w})$ .

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Step 3.3.1.3.1.1.1

Multiply $2$ by $2$ .

$7 w = 4 \sqrt{w} \sqrt{w} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.1.2

Raise $\sqrt{w}$ to the power of $1$ .

$7 w = 4 ({\sqrt{w}}^{1} \sqrt{w}) + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.1.3

Raise $\sqrt{w}$ to the power of $1$ .

$7 w = 4 ({\sqrt{w}}^{1} {\sqrt{w}}^{1}) + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.1.4

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

$7 w = 4 {\sqrt{w}}^{1 + 1} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.1.5

Add $1$ and $1$ .

$7 w = 4 {\sqrt{w}}^{2} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2

Rewrite ${\sqrt{w}}^{2}$ as $w$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{w}$ as $w^{\frac{1}{2}}$ .

$7 w = 4 {(w^{\frac{1}{2}})}^{2} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2.2

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

$7 w = 4 w^{\frac{1}{2} \cdot 2} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2.3

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

$7 w = 4 w^{\frac{2}{2}} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

$7 w = 4 w^{\frac{2}{2}} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

$7 w = 4 w^{1} + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.2.5

Simplify.

$7 w = 4 w + 2 \sqrt{w} \cdot 12 + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.3

Multiply $12$ by $2$ .

$7 w = 4 w + 24 \sqrt{w} + 12 (2 \sqrt{w}) + 12 \cdot 12$

Step 3.3.1.3.1.4

Multiply $2$ by $12$ .

$7 w = 4 w + 24 \sqrt{w} + 24 \sqrt{w} + 12 \cdot 12$

Step 3.3.1.3.1.5

Multiply $12$ by $12$ .

$7 w = 4 w + 24 \sqrt{w} + 24 \sqrt{w} + 144$

Step 3.3.1.3.2

Add $24 \sqrt{w}$ and $24 \sqrt{w}$ .

$7 w = 4 w + 48 \sqrt{w} + 144$

Step 4

Solve for $48 \sqrt{w}$ .

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Step 4.1

Rewrite the equation as $4 w + 48 \sqrt{w} + 144 = 7 w$ .

$4 w + 48 \sqrt{w} + 144 = 7 w$

Step 4.2

Move all terms not containing $48 \sqrt{w}$ to the right side of the equation.

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Step 4.2.1

Subtract $4 w$ from both sides of the equation.

$48 \sqrt{w} + 144 = 7 w - 4 w$

Step 4.2.2

Subtract $144$ from both sides of the equation.

$48 \sqrt{w} = 7 w - 4 w - 144$

Step 4.2.3

Subtract $4 w$ from $7 w$ .

$48 \sqrt{w} = 3 w - 144$

Step 5

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

${(48 \sqrt{w})}^{2} = {(3 w - 144)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{w}$ as $w^{\frac{1}{2}}$ .

${(48 w^{\frac{1}{2}})}^{2} = {(3 w - 144)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(48 w^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $48 w^{\frac{1}{2}}$ .

$48^{2} {(w^{\frac{1}{2}})}^{2} = {(3 w - 144)}^{2}$

Step 6.2.1.2

Raise $48$ to the power of $2$ .

$2304 {(w^{\frac{1}{2}})}^{2} = {(3 w - 144)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${(w^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$2304 w^{\frac{1}{2} \cdot 2} = {(3 w - 144)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$2304 w^{\frac{1}{2} \cdot 2} = {(3 w - 144)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$2304 w^{1} = {(3 w - 144)}^{2}$

Step 6.2.1.4

Simplify.

$2304 w = {(3 w - 144)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(3 w - 144)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(3 w - 144)}^{2}$ as $(3 w - 144) (3 w - 144)$ .

$2304 w = (3 w - 144) (3 w - 144)$

Step 6.3.1.2

Expand $(3 w - 144) (3 w - 144)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$2304 w = 3 w (3 w - 144) - 144 (3 w - 144)$

Step 6.3.1.2.2

Apply the distributive property.

$2304 w = 3 w (3 w) + 3 w \cdot - 144 - 144 (3 w - 144)$

Step 6.3.1.2.3

Apply the distributive property.

$2304 w = 3 w (3 w) + 3 w \cdot - 144 - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$2304 w = 3 \cdot 3 w \cdot w + 3 w \cdot - 144 - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3.1.2

Multiply $w$ by $w$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $w$ .

$2304 w = 3 \cdot 3 (w \cdot w) + 3 w \cdot - 144 - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3.1.2.2

Multiply $w$ by $w$ .

$2304 w = 3 \cdot 3 w^{2} + 3 w \cdot - 144 - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3.1.3

Multiply $3$ by $3$ .

$2304 w = 9 w^{2} + 3 w \cdot - 144 - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3.1.4

Multiply $- 144$ by $3$ .

$2304 w = 9 w^{2} - 432 w - 144 (3 w) - 144 \cdot - 144$

Step 6.3.1.3.1.5

Multiply $3$ by $- 144$ .

$2304 w = 9 w^{2} - 432 w - 432 w - 144 \cdot - 144$

Step 6.3.1.3.1.6

Multiply $- 144$ by $- 144$ .

$2304 w = 9 w^{2} - 432 w - 432 w + 20736$

Step 6.3.1.3.2

Subtract $432 w$ from $- 432 w$ .

$2304 w = 9 w^{2} - 864 w + 20736$

Step 7

Solve for $w$ .

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Step 7.1

Since $w$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 w^{2} - 864 w + 20736 = 2304 w$

Step 7.2

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

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Step 7.2.1

Subtract $2304 w$ from both sides of the equation.

$9 w^{2} - 864 w + 20736 - 2304 w = 0$

Step 7.2.2

Subtract $2304 w$ from $- 864 w$ .

$9 w^{2} - 3168 w + 20736 = 0$

Step 7.3

Factor $9$ out of $9 w^{2} - 3168 w + 20736$ .

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Step 7.3.1

Factor $9$ out of $9 w^{2}$ .

$9 (w^{2}) - 3168 w + 20736 = 0$

Step 7.3.2

Factor $9$ out of $- 3168 w$ .

$9 (w^{2}) + 9 (- 352 w) + 20736 = 0$

Step 7.3.3

Factor $9$ out of $20736$ .

$9 w^{2} + 9 (- 352 w) + 9 \cdot 2304 = 0$

Step 7.3.4

Factor $9$ out of $9 w^{2} + 9 (- 352 w)$ .

$9 (w^{2} - 352 w) + 9 \cdot 2304 = 0$

Step 7.3.5

Factor $9$ out of $9 (w^{2} - 352 w) + 9 \cdot 2304$ .

$9 (w^{2} - 352 w + 2304) = 0$

Step 7.4

Divide each term in $9 (w^{2} - 352 w + 2304) = 0$ by $9$ and simplify.

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Step 7.4.1

Divide each term in $9 (w^{2} - 352 w + 2304) = 0$ by $9$ .

$\frac{9 (w^{2} - 352 w + 2304)}{9} = \frac{0}{9}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $9$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{9 (w^{2} - 352 w + 2304)}{9} = \frac{0}{9}$

Step 7.4.2.1.2

Divide $w^{2} - 352 w + 2304$ by $1$ .

$w^{2} - 352 w + 2304 = \frac{0}{9}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $0$ by $9$ .

$w^{2} - 352 w + 2304 = 0$

Step 7.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.6

Substitute the values $a = 1$ , $b = - 352$ , and $c = 2304$ into the quadratic formula and solve for $w$ .

$\frac{352 \pm \sqrt{{(- 352)}^{2} - 4 \cdot (1 \cdot 2304)}}{2 \cdot 1}$

Step 7.7

Simplify.

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Step 7.7.1

Simplify the numerator.

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Step 7.7.1.1

Raise $- 352$ to the power of $2$ .

$w = \frac{352 \pm \sqrt{123904 - 4 \cdot 1 \cdot 2304}}{2 \cdot 1}$

Step 7.7.1.2

Multiply $- 4 \cdot 1 \cdot 2304$ .

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Step 7.7.1.2.1

Multiply $- 4$ by $1$ .

$w = \frac{352 \pm \sqrt{123904 - 4 \cdot 2304}}{2 \cdot 1}$

Step 7.7.1.2.2

Multiply $- 4$ by $2304$ .

$w = \frac{352 \pm \sqrt{123904 - 9216}}{2 \cdot 1}$

Step 7.7.1.3

Subtract $9216$ from $123904$ .

$w = \frac{352 \pm \sqrt{114688}}{2 \cdot 1}$

Step 7.7.1.4

Rewrite $114688$ as $128^{2} \cdot 7$ .

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Step 7.7.1.4.1

Factor $16384$ out of $114688$ .

$w = \frac{352 \pm \sqrt{16384 (7)}}{2 \cdot 1}$

Step 7.7.1.4.2

Rewrite $16384$ as $128^{2}$ .

$w = \frac{352 \pm \sqrt{128^{2} \cdot 7}}{2 \cdot 1}$

Step 7.7.1.5

Pull terms out from under the radical.

$w = \frac{352 \pm 128 \sqrt{7}}{2 \cdot 1}$

Step 7.7.2

Multiply $2$ by $1$ .

$w = \frac{352 \pm 128 \sqrt{7}}{2}$

Step 7.7.3

Simplify $\frac{352 \pm 128 \sqrt{7}}{2}$ .

$w = 176 \pm 64 \sqrt{7}$

Step 7.8

The final answer is the combination of both solutions.

$w = 176 + 64 \sqrt{7}, 176 - 64 \sqrt{7}$

Step 8

Exclude the solutions that do not make $\sqrt{7 w} - 2 = \sqrt{4 w} + 10$ true.

$w = 176 + 64 \sqrt{7}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$w = 176 + 64 \sqrt{7}$

Decimal Form:

$w = 345.32808390 \dots$