Algebra Examples

$\sqrt{x y + 3 x z} + \sqrt{\frac{y^{2} + y z}{2}}$ , $x + y + z = 3$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 3 - y$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 3 - y - z$

Step 2

Replace the variable $x$ with $3 - y - z$ in the expression.

$\sqrt{(3 - y - z) \cdot y + 3 (3 - y - z) \cdot z} + \sqrt{\frac{y^{2} + y z}{2}}$

Step 3

Simplify each term.

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Step 3.1

Factor $3 - y - z$ out of $(3 - y - z) \cdot y + 3 (3 - y - z) \cdot z$ .

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Step 3.1.1

Factor $3 - y - z$ out of $(3 - y - z) \cdot y$ .

$\sqrt{(3 - y - z) (y) + 3 (3 - y - z) \cdot z} + \sqrt{\frac{y^{2} + y z}{2}}$

Step 3.1.2

Factor $3 - y - z$ out of $3 (3 - y - z) \cdot z$ .

$\sqrt{(3 - y - z) (y) + (3 - y - z) (3 \cdot z)} + \sqrt{\frac{y^{2} + y z}{2}}$

Step 3.1.3

Factor $3 - y - z$ out of $(3 - y - z) (y) + (3 - y - z) (3 \cdot z)$ .

$\sqrt{(3 - y - z) (y + 3 \cdot z)} + \sqrt{\frac{y^{2} + y z}{2}}$

Step 3.2

Factor $y$ out of $y^{2} + y z$ .

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Step 3.2.1

Factor $y$ out of $y^{2}$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \sqrt{\frac{y \cdot y + y z}{2}}$

Step 3.2.2

Factor $y$ out of $y z$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \sqrt{\frac{y \cdot y + y (z)}{2}}$

Step 3.2.3

Factor $y$ out of $y \cdot y + y (z)$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \sqrt{\frac{y (y + z)}{2}}$

Step 3.3

Rewrite $\sqrt{\frac{y (y + z)}{2}}$ as $\frac{\sqrt{y (y + z)}}{\sqrt{2}}$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)}}{\sqrt{2}}$

Step 3.4

Multiply $\frac{\sqrt{y (y + z)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{y (y + z)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.5.4

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

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.5.6

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

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Step 3.5.6.1

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

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{2^{1}}$

Step 3.5.6.5

Evaluate the exponent.

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z)} \sqrt{2}}{2}$

Step 3.6

Combine using the product rule for radicals.

$\sqrt{(3 - y - z) (y + 3 z)} + \frac{\sqrt{y (y + z) \cdot 2}}{2}$

Step 4

To write $\sqrt{(3 - y - z) (y + 3 z)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{(3 - y - z) (y + 3 z)} \cdot \frac{2}{2} + \frac{\sqrt{y (y + z) \cdot 2}}{2}$

Step 5

Simplify terms.

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Step 5.1

Combine $\sqrt{(3 - y - z) (y + 3 z)}$ and $\frac{2}{2}$ .

$\frac{\sqrt{(3 - y - z) (y + 3 z)} \cdot 2}{2} + \frac{\sqrt{y (y + z) \cdot 2}}{2}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{\sqrt{(3 - y - z) (y + 3 z)} \cdot 2 + \sqrt{y (y + z) \cdot 2}}{2}$

Step 6

Move $2$ to the left of $\sqrt{(3 - y - z) (y + 3 z)}$ .

$\frac{2 \sqrt{(3 - y - z) (y + 3 z)} + \sqrt{y (y + z) \cdot 2}}{2}$

Step 7

Reorder factors in $\frac{2 \sqrt{(3 - y - z) (y + 3 z)} + \sqrt{y (y + z) \cdot 2}}{2}$ .

$\frac{2 \sqrt{(3 - y - z) (y + 3 z)} + \sqrt{2 y (y + z)}}{2}$