Basic Math Examples

$\sqrt{16 y + 1} = 2 \sqrt{7 y} - 1$

Step 1

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

${\sqrt{16 y + 1}}^{2} = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 y + 1}$ as ${(16 y + 1)}^{\frac{1}{2}}$ .

${({(16 y + 1)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(16 y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(16 y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(16 y + 1)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(16 y + 1)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(16 y + 1)}^{1} = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2.2.1.2

Simplify.

$16 y + 1 = {(2 \sqrt{7 y} - 1)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 \sqrt{7 y} - 1)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(2 \sqrt{7 y} - 1)}^{2}$ as $(2 \sqrt{7 y} - 1) (2 \sqrt{7 y} - 1)$ .

$16 y + 1 = (2 \sqrt{7 y} - 1) (2 \sqrt{7 y} - 1)$

Step 2.3.1.2

Expand $(2 \sqrt{7 y} - 1) (2 \sqrt{7 y} - 1)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$16 y + 1 = 2 \sqrt{7 y} (2 \sqrt{7 y} - 1) - 1 (2 \sqrt{7 y} - 1)$

Step 2.3.1.2.2

Apply the distributive property.

$16 y + 1 = 2 \sqrt{7 y} (2 \sqrt{7 y}) + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y} - 1)$

Step 2.3.1.2.3

Apply the distributive property.

$16 y + 1 = 2 \sqrt{7 y} (2 \sqrt{7 y}) + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $2 \sqrt{7 y} (2 \sqrt{7 y})$ .

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Step 2.3.1.3.1.1.1

Multiply $2$ by $2$ .

$16 y + 1 = 4 \sqrt{7 y} \sqrt{7 y} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.1.2

Raise $\sqrt{7 y}$ to the power of $1$ .

$16 y + 1 = 4 ({\sqrt{7 y}}^{1} \sqrt{7 y}) + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.1.3

Raise $\sqrt{7 y}$ to the power of $1$ .

$16 y + 1 = 4 ({\sqrt{7 y}}^{1} {\sqrt{7 y}}^{1}) + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.1.4

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

$16 y + 1 = 4 {\sqrt{7 y}}^{1 + 1} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.1.5

Add $1$ and $1$ .

$16 y + 1 = 4 {\sqrt{7 y}}^{2} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2

Rewrite ${\sqrt{7 y}}^{2}$ as $7 y$ .

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Step 2.3.1.3.1.2.1

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

$16 y + 1 = 4 {({(7 y)}^{\frac{1}{2}})}^{2} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2.2

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

$16 y + 1 = 4 {(7 y)}^{\frac{1}{2} \cdot 2} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2.3

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

$16 y + 1 = 4 {(7 y)}^{\frac{2}{2}} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

$16 y + 1 = 4 {(7 y)}^{\frac{2}{2}} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$16 y + 1 = 4 {(7 y)}^{1} + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.2.5

Simplify.

$16 y + 1 = 4 (7 y) + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.3

Multiply $7$ by $4$ .

$16 y + 1 = 28 y + 2 \sqrt{7 y} \cdot - 1 - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.4

Multiply $- 1$ by $2$ .

$16 y + 1 = 28 y - 2 \sqrt{7 y} - 1 (2 \sqrt{7 y}) - 1 \cdot - 1$

Step 2.3.1.3.1.5

Multiply $2$ by $- 1$ .

$16 y + 1 = 28 y - 2 \sqrt{7 y} - 2 \sqrt{7 y} - 1 \cdot - 1$

Step 2.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$16 y + 1 = 28 y - 2 \sqrt{7 y} - 2 \sqrt{7 y} + 1$

Step 2.3.1.3.2

Subtract $2 \sqrt{7 y}$ from $- 2 \sqrt{7 y}$ .

$16 y + 1 = 28 y - 4 \sqrt{7 y} + 1$

Step 3

Solve for $- 4 \sqrt{7 y}$ .

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Step 3.1

Rewrite the equation as $28 y - 4 \sqrt{7 y} + 1 = 16 y + 1$ .

$28 y - 4 \sqrt{7 y} + 1 = 16 y + 1$

Step 3.2

Move all terms not containing $- 4 \sqrt{7 y}$ to the right side of the equation.

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Step 3.2.1

Subtract $28 y$ from both sides of the equation.

$- 4 \sqrt{7 y} + 1 = 16 y + 1 - 28 y$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$- 4 \sqrt{7 y} = 16 y + 1 - 28 y - 1$

Step 3.2.3

Combine the opposite terms in $16 y + 1 - 28 y - 1$ .

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Step 3.2.3.1

Subtract $1$ from $1$ .

$- 4 \sqrt{7 y} = 16 y - 28 y + 0$

Step 3.2.3.2

Add $16 y - 28 y$ and $0$ .

$- 4 \sqrt{7 y} = 16 y - 28 y$

Step 3.2.4

Subtract $28 y$ from $16 y$ .

$- 4 \sqrt{7 y} = - 12 y$

Step 4

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

${(- 4 \sqrt{7 y})}^{2} = {(- 12 y)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${(- 4 {(7 y)}^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 4 {(7 y)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $7 y$ .

${(- 4 (7^{\frac{1}{2}} y^{\frac{1}{2}}))}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.2.1

Apply the product rule to $- 4 \cdot 7^{\frac{1}{2}} y^{\frac{1}{2}}$ .

${(- 4 \cdot 7^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.2.2

Apply the product rule to $- 4 \cdot 7^{\frac{1}{2}}$ .

${(- 4)}^{2} \cdot {(7^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.3

Raise $- 4$ to the power of $2$ .

$16 \cdot {(7^{\frac{1}{2}})}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.4

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

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Step 5.2.1.4.1

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

$16 \cdot 7^{\frac{1}{2} \cdot 2} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.4.2

Cancel the common factor of $2$ .

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Step 5.2.1.4.2.1

Cancel the common factor.

$16 \cdot 7^{\frac{1}{2} \cdot 2} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.4.2.2

Rewrite the expression.

$16 \cdot 7^{1} {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.5

Evaluate the exponent.

$16 \cdot 7 {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.6

Multiply $16$ by $7$ .

$112 {(y^{\frac{1}{2}})}^{2} = {(- 12 y)}^{2}$

Step 5.2.1.7

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

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Step 5.2.1.7.1

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

$112 y^{\frac{1}{2} \cdot 2} = {(- 12 y)}^{2}$

Step 5.2.1.7.2

Cancel the common factor of $2$ .

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Step 5.2.1.7.2.1

Cancel the common factor.

$112 y^{\frac{1}{2} \cdot 2} = {(- 12 y)}^{2}$

Step 5.2.1.7.2.2

Rewrite the expression.

$112 y^{1} = {(- 12 y)}^{2}$

Step 5.2.1.8

Simplify.

$112 y = {(- 12 y)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(- 12 y)}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $- 12 y$ .

$112 y = {(- 12)}^{2} y^{2}$

Step 5.3.1.2

Raise $- 12$ to the power of $2$ .

$112 y = 144 y^{2}$

Step 6

Solve for $y$ .

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Step 6.1

Subtract $144 y^{2}$ from both sides of the equation.

$112 y - 144 y^{2} = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$112 u - 144 u^{2} = 0$

Step 6.2.2

Factor $16 u$ out of $112 u - 144 u^{2}$ .

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Step 6.2.2.1

Factor $16 u$ out of $112 u$ .

$16 u (7) - 144 u^{2} = 0$

Step 6.2.2.2

Factor $16 u$ out of $- 144 u^{2}$ .

$16 u (7) + 16 u (- 9 u) = 0$

Step 6.2.2.3

Factor $16 u$ out of $16 u (7) + 16 u (- 9 u)$ .

$16 u (7 - 9 u) = 0$

Step 6.2.3

Replace all occurrences of $u$ with $y$ .

$16 y (7 - 9 y) = 0$

Step 6.3

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

$y = 0$

$7 - 9 y = 0$

Step 6.4

Set $y$ equal to $0$ .

$y = 0$

Step 6.5

Set $7 - 9 y$ equal to $0$ and solve for $y$ .

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Step 6.5.1

Set $7 - 9 y$ equal to $0$ .

$7 - 9 y = 0$

Step 6.5.2

Solve $7 - 9 y = 0$ for $y$ .

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Step 6.5.2.1

Subtract $7$ from both sides of the equation.

$- 9 y = - 7$

Step 6.5.2.2

Divide each term in $- 9 y = - 7$ by $- 9$ and simplify.

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Step 6.5.2.2.1

Divide each term in $- 9 y = - 7$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{- 7}{- 9}$

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Cancel the common factor of $- 9$ .

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Step 6.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- 7}{- 9}$

Step 6.5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 9}$

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{7}{9}$

Step 6.6

The final solution is all the values that make $16 y (7 - 9 y) = 0$ true.

$y = 0, \frac{7}{9}$

Step 7

Exclude the solutions that do not make $\sqrt{16 y + 1} = 2 \sqrt{7 y} - 1$ true.

$y = \frac{7}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \frac{7}{9}$

Decimal Form:

$y = 0.\overline{7}$