Finite Math Examples

$419 \sqrt{32} x = 1084$

Step 1

Simplify.

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Step 1.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.1.1

Factor $16$ out of $32$ .

$419 \sqrt{16 (2)} x = 1084$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

$419 \sqrt{4^{2} \cdot 2} x = 1084$

Step 1.2

Pull terms out from under the radical.

$419 (4 \sqrt{2}) x = 1084$

Step 1.3

Multiply $4$ by $419$ .

$1676 \sqrt{2} x = 1084$

Step 2

Divide each term in $1676 \sqrt{2} x = 1084$ by $1676 \sqrt{2}$ and simplify.

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Step 2.1

Divide each term in $1676 \sqrt{2} x = 1084$ by $1676 \sqrt{2}$ .

$\frac{1676 \sqrt{2} x}{1676 \sqrt{2}} = \frac{1084}{1676 \sqrt{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $1676$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{1676 \sqrt{2} x}{1676 \sqrt{2}} = \frac{1084}{1676 \sqrt{2}}$

Step 2.2.1.2

Rewrite the expression.

$\frac{\sqrt{2} x}{\sqrt{2}} = \frac{1084}{1676 \sqrt{2}}$

Step 2.2.2

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

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Step 2.2.2.1

Cancel the common factor.

$\frac{\sqrt{2} x}{\sqrt{2}} = \frac{1084}{1676 \sqrt{2}}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x = \frac{1084}{1676 \sqrt{2}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $1084$ and $1676$ .

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Step 2.3.1.1

Factor $4$ out of $1084$ .

$x = \frac{4 (271)}{1676 \sqrt{2}}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $4$ out of $1676 \sqrt{2}$ .

$x = \frac{4 (271)}{4 (419 \sqrt{2})}$

Step 2.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 271}{4 (419 \sqrt{2})}$

Step 2.3.1.2.3

Rewrite the expression.

$x = \frac{271}{419 \sqrt{2}}$

Step 2.3.2

Multiply $\frac{271}{419 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{271}{419 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.3.3

Combine and simplify the denominator.

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Step 2.3.3.1

Multiply $\frac{271}{419 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{271 \sqrt{2}}{419 \sqrt{2} \sqrt{2}}$

Step 2.3.3.2

Move $\sqrt{2}$ .

$x = \frac{271 \sqrt{2}}{419 (\sqrt{2} \sqrt{2})}$

Step 2.3.3.3

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

$x = \frac{271 \sqrt{2}}{419 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 2.3.3.4

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

$x = \frac{271 \sqrt{2}}{419 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 2.3.3.5

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

$x = \frac{271 \sqrt{2}}{419 {\sqrt{2}}^{1 + 1}}$

Step 2.3.3.6

Add $1$ and $1$ .

$x = \frac{271 \sqrt{2}}{419 {\sqrt{2}}^{2}}$

Step 2.3.3.7

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

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Step 2.3.3.7.1

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

$x = \frac{271 \sqrt{2}}{419 {(2^{\frac{1}{2}})}^{2}}$

Step 2.3.3.7.2

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

$x = \frac{271 \sqrt{2}}{419 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.3.3.7.3

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

$x = \frac{271 \sqrt{2}}{419 \cdot 2^{\frac{2}{2}}}$

Step 2.3.3.7.4

Cancel the common factor of $2$ .

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Step 2.3.3.7.4.1

Cancel the common factor.

$x = \frac{271 \sqrt{2}}{419 \cdot 2^{\frac{2}{2}}}$

Step 2.3.3.7.4.2

Rewrite the expression.

$x = \frac{271 \sqrt{2}}{419 \cdot 2^{1}}$

Step 2.3.3.7.5

Evaluate the exponent.

$x = \frac{271 \sqrt{2}}{419 \cdot 2}$

Step 2.3.4

Multiply $419$ by $2$ .

$x = \frac{271 \sqrt{2}}{838}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{271 \sqrt{2}}{838}$

Decimal Form:

$x = 0.45734114 \dots$