Algebra Examples

$\frac{\sqrt{2}}{3} r + 1 = 23$

Step 1

Combine $\frac{\sqrt{2}}{3}$ and $r$ .

$\frac{\sqrt{2} r}{3} + 1 = 23$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

$\frac{\sqrt{2} r}{3} = 23 - 1$

Step 2.2

Subtract $1$ from $23$ .

$\frac{\sqrt{2} r}{3} = 22$

Step 3

Multiply both sides of the equation by $\frac{3}{\sqrt{2}}$ .

$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2} r}{3} = \frac{3}{\sqrt{2}} \cdot 22$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2} r}{3}$ .

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2} r}{3} = \frac{3}{\sqrt{2}} \cdot 22$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{\sqrt{2}} (\sqrt{2} r) = \frac{3}{\sqrt{2}} \cdot 22$

Step 4.1.1.2

Cancel the common factor of $\sqrt{2}$ .

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Step 4.1.1.2.1

Factor $\sqrt{2}$ out of $\sqrt{2} r$ .

$\frac{1}{\sqrt{2}} (\sqrt{2} (r)) = \frac{3}{\sqrt{2}} \cdot 22$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{2}} (\sqrt{2} r) = \frac{3}{\sqrt{2}} \cdot 22$

Step 4.1.1.2.3

Rewrite the expression.

$r = \frac{3}{\sqrt{2}} \cdot 22$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{\sqrt{2}} \cdot 22$ .

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Step 4.2.1.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot 22$

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$r = \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot 22$

Step 4.2.1.2.2

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

$r = \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cdot 22$

Step 4.2.1.2.3

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

$r = \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cdot 22$

Step 4.2.1.2.4

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

$r = \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot 22$

Step 4.2.1.2.5

Add $1$ and $1$ .

$r = \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}} \cdot 22$

Step 4.2.1.2.6

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

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Step 4.2.1.2.6.1

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

$r = \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot 22$

Step 4.2.1.2.6.2

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

$r = \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot 22$

Step 4.2.1.2.6.3

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

$r = \frac{3 \sqrt{2}}{2^{\frac{2}{2}}} \cdot 22$

Step 4.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.6.4.1

Cancel the common factor.

$r = \frac{3 \sqrt{2}}{2^{\frac{2}{2}}} \cdot 22$

Step 4.2.1.2.6.4.2

Rewrite the expression.

$r = \frac{3 \sqrt{2}}{2^{1}} \cdot 22$

Step 4.2.1.2.6.5

Evaluate the exponent.

$r = \frac{3 \sqrt{2}}{2} \cdot 22$

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Factor $2$ out of $22$ .

$r = \frac{3 \sqrt{2}}{2} \cdot (2 (11))$

Step 4.2.1.3.2

Cancel the common factor.

$r = \frac{3 \sqrt{2}}{2} \cdot (2 \cdot 11)$

Step 4.2.1.3.3

Rewrite the expression.

$r = 3 \sqrt{2} \cdot 11$

Step 4.2.1.4

Multiply $11$ by $3$ .

$r = 33 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = 33 \sqrt{2}$

Decimal Form:

$r = 46.66904755 \dots$