Basic Math Examples

$329 = \frac{k^{2} \cdot (2400 {(35)}^{2})}{600}$

Step 1

Rewrite the equation as $\frac{k^{2} \cdot 2400 \cdot 35^{2}}{600} = 329$ .

$\frac{k^{2} \cdot 2400 \cdot 35^{2}}{600} = 329$

Step 2

Cancel the common factor of $2400$ and $600$ .

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Step 2.1

Factor $600$ out of $k^{2} \cdot 2400 \cdot 35^{2}$ .

$\frac{600 (k^{2} \cdot 4 \cdot 35^{2})}{600} = 329$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $600$ out of $600$ .

$\frac{600 (k^{2} \cdot 4 \cdot 35^{2})}{600 (1)} = 329$

Step 2.2.2

Cancel the common factor.

$\frac{600 (k^{2} \cdot 4 \cdot 35^{2})}{600 \cdot 1} = 329$

Step 2.2.3

Rewrite the expression.

$\frac{k^{2} \cdot 4 \cdot 35^{2}}{1} = 329$

Step 2.2.4

Divide $k^{2} \cdot 4 \cdot 35^{2}$ by $1$ .

$k^{2} \cdot 4 \cdot 35^{2} = 329$

Step 3

Solve for $k$ .

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Step 3.1

Divide each term in $k^{2} \cdot 4 \cdot 35^{2} = 329$ by $4900$ and simplify.

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Step 3.1.1

Divide each term in $k^{2} \cdot 4 \cdot 35^{2} = 329$ by $4900$ .

$\frac{k^{2} \cdot 4 \cdot 35^{2}}{4900} = \frac{329}{4900}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $4$ and $4900$ .

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Step 3.1.2.1.1

Factor $4$ out of $k^{2} \cdot 4 \cdot 35^{2}$ .

$\frac{4 (k^{2} \cdot 35^{2})}{4900} = \frac{329}{4900}$

Step 3.1.2.1.2

Cancel the common factors.

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Step 3.1.2.1.2.1

Factor $4$ out of $4900$ .

$\frac{4 (k^{2} \cdot 35^{2})}{4 \cdot 1225} = \frac{329}{4900}$

Step 3.1.2.1.2.2

Cancel the common factor.

$\frac{4 (k^{2} \cdot 35^{2})}{4 \cdot 1225} = \frac{329}{4900}$

Step 3.1.2.1.2.3

Rewrite the expression.

$\frac{k^{2} \cdot 35^{2}}{1225} = \frac{329}{4900}$

Step 3.1.2.2

Raise $35$ to the power of $2$ .

$\frac{k^{2} \cdot 1225}{1225} = \frac{329}{4900}$

Step 3.1.2.3

Cancel the common factor of $1225$ .

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Step 3.1.2.3.1

Cancel the common factor.

$\frac{k^{2} \cdot 1225}{1225} = \frac{329}{4900}$

Step 3.1.2.3.2

Divide $k^{2}$ by $1$ .

$k^{2} = \frac{329}{4900}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $329$ and $4900$ .

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Step 3.1.3.1.1

Factor $7$ out of $329$ .

$k^{2} = \frac{7 (47)}{4900}$

Step 3.1.3.1.2

Cancel the common factors.

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Step 3.1.3.1.2.1

Factor $7$ out of $4900$ .

$k^{2} = \frac{7 \cdot 47}{7 \cdot 700}$

Step 3.1.3.1.2.2

Cancel the common factor.

$k^{2} = \frac{7 \cdot 47}{7 \cdot 700}$

Step 3.1.3.1.2.3

Rewrite the expression.

$k^{2} = \frac{47}{700}$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$k = \pm \sqrt{\frac{47}{700}}$

Step 3.3

Simplify $\pm \sqrt{\frac{47}{700}}$ .

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Step 3.3.1

Rewrite $\sqrt{\frac{47}{700}}$ as $\frac{\sqrt{47}}{\sqrt{700}}$ .

$k = \pm \frac{\sqrt{47}}{\sqrt{700}}$

Step 3.3.2

Simplify the denominator.

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Step 3.3.2.1

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 3.3.2.1.1

Factor $100$ out of $700$ .

$k = \pm \frac{\sqrt{47}}{\sqrt{100 (7)}}$

Step 3.3.2.1.2

Rewrite $100$ as $10^{2}$ .

$k = \pm \frac{\sqrt{47}}{\sqrt{10^{2} \cdot 7}}$

Step 3.3.2.2

Pull terms out from under the radical.

$k = \pm \frac{\sqrt{47}}{10 \sqrt{7}}$

Step 3.3.3

Multiply $\frac{\sqrt{47}}{10 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$k = \pm \frac{\sqrt{47}}{10 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3.3.4

Combine and simplify the denominator.

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Step 3.3.4.1

Multiply $\frac{\sqrt{47}}{10 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \sqrt{7} \sqrt{7}}$

Step 3.3.4.2

Move $\sqrt{7}$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 (\sqrt{7} \sqrt{7})}$

Step 3.3.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 ({\sqrt{7}}^{1} \sqrt{7})}$

Step 3.3.4.4

Raise $\sqrt{7}$ to the power of $1$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}$

Step 3.3.4.5

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

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 {\sqrt{7}}^{1 + 1}}$

Step 3.3.4.6

Add $1$ and $1$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 {\sqrt{7}}^{2}}$

Step 3.3.4.7

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 3.3.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 {(7^{\frac{1}{2}})}^{2}}$

Step 3.3.4.7.2

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

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \cdot 7^{\frac{1}{2} \cdot 2}}$

Step 3.3.4.7.3

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

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \cdot 7^{\frac{2}{2}}}$

Step 3.3.4.7.4

Cancel the common factor of $2$ .

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Step 3.3.4.7.4.1

Cancel the common factor.

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \cdot 7^{\frac{2}{2}}}$

Step 3.3.4.7.4.2

Rewrite the expression.

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \cdot 7^{1}}$

Step 3.3.4.7.5

Evaluate the exponent.

$k = \pm \frac{\sqrt{47} \sqrt{7}}{10 \cdot 7}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Combine using the product rule for radicals.

$k = \pm \frac{\sqrt{47 \cdot 7}}{10 \cdot 7}$

Step 3.3.5.2

Multiply $47$ by $7$ .

$k = \pm \frac{\sqrt{329}}{10 \cdot 7}$

Step 3.3.6

Multiply $10$ by $7$ .

$k = \pm \frac{\sqrt{329}}{70}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$k = \frac{\sqrt{329}}{70}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$k = - \frac{\sqrt{329}}{70}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$k = \frac{\sqrt{329}}{70}, - \frac{\sqrt{329}}{70}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$k = \frac{\sqrt{329}}{70}, - \frac{\sqrt{329}}{70}$

Decimal Form:

$k = 0.25911938 \dots, - 0.25911938 \dots$