Algebra Examples

$(\sqrt{x} - y + 1) (\sqrt{x} + y - 1)$

Step 1

Expand $(\sqrt{x} - y + 1) (\sqrt{x} + y - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{x} \sqrt{x} + \sqrt{x} y + \sqrt{x} \cdot - 1 - y \sqrt{x} - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{x} \sqrt{x} + \sqrt{x} y + \sqrt{x} \cdot - 1 - y \sqrt{x} - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{x} y$ and $- y \sqrt{x}$ .

$\sqrt{x} \sqrt{x} + y \sqrt{x} + \sqrt{x} \cdot - 1 - y \sqrt{x} - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.1.2

Subtract $y \sqrt{x}$ from $y \sqrt{x}$ .

$\sqrt{x} \sqrt{x} + 0 + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.1.3

Add $\sqrt{x} \sqrt{x}$ and $0$ .

$\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 2.2.1.1

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

${\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.1.2

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

${\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.1.3

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

${\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.1.4

Add $1$ and $1$ .

${\sqrt{x}}^{2} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

$x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.2.3

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

$x^{\frac{2}{2}} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$x^{\frac{2}{2}} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.2.4.2

Rewrite the expression.

$x^{1} + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.2.5

Simplify.

$x + \sqrt{x} \cdot - 1 - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.3

Move $- 1$ to the left of $\sqrt{x}$ .

$x - 1 \cdot \sqrt{x} - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.4

Rewrite $- 1 \sqrt{x}$ as $- \sqrt{x}$ .

$x - \sqrt{x} - y \cdot y - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.5.1

Move $y$ .

$x - \sqrt{x} - (y \cdot y) - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.5.2

Multiply $y$ by $y$ .

$x - \sqrt{x} - y^{2} - y \cdot - 1 + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.6

Multiply $- y \cdot - 1$ .

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Step 2.2.6.1

Multiply $- 1$ by $- 1$ .

$x - \sqrt{x} - y^{2} + 1 y + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.6.2

Multiply $y$ by $1$ .

$x - \sqrt{x} - y^{2} + y + 1 \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.7

Multiply $\sqrt{x}$ by $1$ .

$x - \sqrt{x} - y^{2} + y + \sqrt{x} + 1 y + 1 \cdot - 1$

Step 2.2.8

Multiply $y$ by $1$ .

$x - \sqrt{x} - y^{2} + y + \sqrt{x} + y + 1 \cdot - 1$

Step 2.2.9

Multiply $- 1$ by $1$ .

$x - \sqrt{x} - y^{2} + y + \sqrt{x} + y - 1$

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Combine the opposite terms in $x - \sqrt{x} - y^{2} + y + \sqrt{x} + y - 1$ .

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Step 2.3.1.1

Add $- \sqrt{x}$ and $\sqrt{x}$ .

$x - y^{2} + y + 0 + y - 1$

Step 2.3.1.2

Add $x - y^{2} + y$ and $0$ .

$x - y^{2} + y + y - 1$

Step 2.3.2

Add $y$ and $y$ .

$x - y^{2} + 2 y - 1$

Step 2.3.3

Reorder $x$ and $- y^{2}$ .

$- y^{2} + x + 2 y - 1$