Basic Math Examples

$5 \sqrt{y} = y$

Step 1

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

${(5 \sqrt{y})}^{2} = y^{2}$

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 $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

${(5 y^{\frac{1}{2}})}^{2} = y^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(5 y^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $5 y^{\frac{1}{2}}$ .

$5^{2} {(y^{\frac{1}{2}})}^{2} = y^{2}$

Step 2.2.1.2

Raise $5$ to the power of $2$ .

$25 {(y^{\frac{1}{2}})}^{2} = y^{2}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$25 y^{\frac{1}{2} \cdot 2} = y^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$25 y^{\frac{1}{2} \cdot 2} = y^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$25 y^{1} = y^{2}$

Step 2.2.1.4

Simplify.

$25 y = y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

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

$25 y - y^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

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

$25 u - u^{2} = 0$

Step 3.2.2

Factor $u$ out of $25 u - u^{2}$ .

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Step 3.2.2.1

Factor $u$ out of $25 u$ .

$u \cdot 25 - u^{2} = 0$

Step 3.2.2.2

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

$u \cdot 25 + u (- u) = 0$

Step 3.2.2.3

Factor $u$ out of $u \cdot 25 + u (- u)$ .

$u (25 - u) = 0$

Step 3.2.3

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

$y (25 - y) = 0$

Step 3.3

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

$y = 0$

$25 - y = 0$

Step 3.4

Set $y$ equal to $0$ .

$y = 0$

Step 3.5

Set $25 - y$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $25 - y$ equal to $0$ .

$25 - y = 0$

Step 3.5.2

Solve $25 - y = 0$ for $y$ .

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Step 3.5.2.1

Subtract $25$ from both sides of the equation.

$- y = - 25$

Step 3.5.2.2

Divide each term in $- y = - 25$ by $- 1$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- y = - 25$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 25}{- 1}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 25}{- 1}$

Step 3.5.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 25}{- 1}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Divide $- 25$ by $- 1$ .

$y = 25$

Step 3.6

The final solution is all the values that make $y (25 - y) = 0$ true.

$y = 0, 25$