Algebra Examples

$10 \sqrt{\frac{x}{5}} = 2 \sqrt{5 x}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${(10 \sqrt{\frac{x}{5}})}^{2} = {(2 \sqrt{5 x})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x}{5}}$ as ${(\frac{x}{5})}^{\frac{1}{2}}$ .

${(10 {(\frac{x}{5})}^{\frac{1}{2}})}^{2} = {(2 \sqrt{5 x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(10 {(\frac{x}{5})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $\frac{x}{5}$ .

${(10 \frac{x^{\frac{1}{2}}}{5^{\frac{1}{2}}})}^{2} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.2

Combine $10$ and $\frac{x^{\frac{1}{2}}}{5^{\frac{1}{2}}}$ .

${(\frac{10 x^{\frac{1}{2}}}{5^{\frac{1}{2}}})}^{2} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.3.1

Apply the product rule to $\frac{10 x^{\frac{1}{2}}}{5^{\frac{1}{2}}}$ .

$\frac{{(10 x^{\frac{1}{2}})}^{2}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.3.2

Apply the product rule to $10 x^{\frac{1}{2}}$ .

$\frac{10^{2} {(x^{\frac{1}{2}})}^{2}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Raise $10$ to the power of $2$ .

$\frac{100 {(x^{\frac{1}{2}})}^{2}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.4.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.4.2.1

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

$\frac{100 x^{\frac{1}{2} \cdot 2}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.4.2.2

Cancel the common factor of $2$ .

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Step 2.2.1.4.2.2.1

Cancel the common factor.

$\frac{100 x^{\frac{1}{2} \cdot 2}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.4.2.2.2

Rewrite the expression.

$\frac{100 x^{1}}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.4.3

Simplify.

$\frac{100 x}{{(5^{\frac{1}{2}})}^{2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.5

Simplify the denominator.

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Step 2.2.1.5.1

Multiply the exponents in ${(5^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.5.1.1

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

$\frac{100 x}{5^{\frac{1}{2} \cdot 2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.5.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.5.1.2.1

Cancel the common factor.

$\frac{100 x}{5^{\frac{1}{2} \cdot 2}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.5.1.2.2

Rewrite the expression.

$\frac{100 x}{5^{1}} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.5.2

Evaluate the exponent.

$\frac{100 x}{5} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.6

Cancel the common factor of $100$ and $5$ .

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Step 2.2.1.6.1

Factor $5$ out of $100 x$ .

$\frac{5 (20 x)}{5} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.6.2

Cancel the common factors.

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Step 2.2.1.6.2.1

Factor $5$ out of $5$ .

$\frac{5 (20 x)}{5 (1)} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.6.2.2

Cancel the common factor.

$\frac{5 (20 x)}{5 \cdot 1} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.6.2.3

Rewrite the expression.

$\frac{20 x}{1} = {(2 \sqrt{5 x})}^{2}$

Step 2.2.1.6.2.4

Divide $20 x$ by $1$ .

$20 x = {(2 \sqrt{5 x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(2 \sqrt{5 x})}^{2}$ .

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Step 2.3.1.1

Simplify the expression.

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Step 2.3.1.1.1

Apply the product rule to $2 \sqrt{5 x}$ .

$20 x = 2^{2} {\sqrt{5 x}}^{2}$

Step 2.3.1.1.2

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

$20 x = 4 {\sqrt{5 x}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{5 x}}^{2}$ as $5 x$ .

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$20 x = 4 {({(5 x)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$20 x = 4 {(5 x)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$20 x = 4 {(5 x)}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$20 x = 4 {(5 x)}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$20 x = 4 {(5 x)}^{1}$

Step 2.3.1.2.5

Simplify.

$20 x = 4 (5 x)$

Step 2.3.1.3

Multiply $5$ by $4$ .

$20 x = 20 x$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $20 x$ from both sides of the equation.

$20 x - 20 x = 0$

Step 3.1.2

Subtract $20 x$ from $20 x$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$