Basic Math Examples

${(2 \sqrt{10})}^{2} + 6 {(\sqrt{10})}^{2} = d^{2}$

Step 1

Rewrite the equation as $d^{2} = {(2 \sqrt{10})}^{2} + 6 {\sqrt{10}}^{2}$ .

$d^{2} = {(2 \sqrt{10})}^{2} + 6 {\sqrt{10}}^{2}$

Step 2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$d^{2} = {(2 \sqrt{10})}^{2} + 6 {(10^{\frac{1}{2}})}^{2}$

Step 2.2

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

$d^{2} = {(2 \sqrt{10})}^{2} + 6 \cdot 10^{\frac{1}{2} \cdot 2}$

Step 2.3

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

$d^{2} = {(2 \sqrt{10})}^{2} + 6 \cdot 10^{\frac{2}{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$d^{2} = {(2 \sqrt{10})}^{2} + 6 \cdot 10^{\frac{2}{2}}$

Step 2.4.2

Rewrite the expression.

$d^{2} = {(2 \sqrt{10})}^{2} + 6 \cdot 10^{1}$

Step 2.5

Evaluate the exponent.

$d^{2} = {(2 \sqrt{10})}^{2} + 6 \cdot 10$

Step 3

Multiply $6$ by $10$ .

$d^{2} = {(2 \sqrt{10})}^{2} + 60$

Step 4

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

$d = \pm \sqrt{{(2 \sqrt{10})}^{2} + 60}$

Step 5

Simplify $\pm \sqrt{{(2 \sqrt{10})}^{2} + 60}$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Apply the product rule to $2 \sqrt{10}$ .

$d = \pm \sqrt{2^{2} {\sqrt{10}}^{2} + 60}$

Step 5.1.2

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

$d = \pm \sqrt{4 {\sqrt{10}}^{2} + 60}$

Step 5.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$d = \pm \sqrt{4 {(10^{\frac{1}{2}})}^{2} + 60}$

Step 5.2.2

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

$d = \pm \sqrt{4 \cdot 10^{\frac{1}{2} \cdot 2} + 60}$

Step 5.2.3

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

$d = \pm \sqrt{4 \cdot 10^{\frac{2}{2}} + 60}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$d = \pm \sqrt{4 \cdot 10^{\frac{2}{2}} + 60}$

Step 5.2.4.2

Rewrite the expression.

$d = \pm \sqrt{4 \cdot 10^{1} + 60}$

Step 5.2.5

Evaluate the exponent.

$d = \pm \sqrt{4 \cdot 10 + 60}$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $4$ by $10$ .

$d = \pm \sqrt{40 + 60}$

Step 5.3.2

Add $40$ and $60$ .

$d = \pm \sqrt{100}$

Step 5.3.3

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

$d = \pm \sqrt{10^{2}}$

Step 5.3.4

Pull terms out from under the radical, assuming positive real numbers.

$d = \pm 10$

Step 6

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

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Step 6.1

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

$d = 10$

Step 6.2

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

$d = - 10$

Step 6.3

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

$d = 10, - 10$