Algebra Examples

$\sqrt{1 \frac{4}{5}} \cdot \sqrt{0.2}$

Step 1

Convert $1 \frac{4}{5}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\sqrt{1 + \frac{4}{5}} \cdot \sqrt{0.2}$

Step 1.2

Add $1$ and $\frac{4}{5}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{5}{5} + \frac{4}{5}} \cdot \sqrt{0.2}$

Step 1.2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + 4}{5}} \cdot \sqrt{0.2}$

Step 1.2.3

Add $5$ and $4$ .

$\sqrt{\frac{9}{5}} \cdot \sqrt{0.2}$

Step 2

Rewrite $\sqrt{\frac{9}{5}}$ as $\frac{\sqrt{9}}{\sqrt{5}}$ .

$\frac{\sqrt{9}}{\sqrt{5}} \cdot \sqrt{0.2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}}}{\sqrt{5}} \cdot \sqrt{0.2}$

Step 3.2

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

$\frac{3}{\sqrt{5}} \cdot \sqrt{0.2}$

Step 4

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \cdot \sqrt{0.2}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}} \cdot \sqrt{0.2}$

Step 5.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \cdot \sqrt{0.2}$

Step 5.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \cdot \sqrt{0.2}$

Step 5.4

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

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \cdot \sqrt{0.2}$

Step 5.5

Add $1$ and $1$ .

$\frac{3 \sqrt{5}}{{\sqrt{5}}^{2}} \cdot \sqrt{0.2}$

Step 5.6

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

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Step 5.6.1

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

$\frac{3 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \cdot \sqrt{0.2}$

Step 5.6.2

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

$\frac{3 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \cdot \sqrt{0.2}$

Step 5.6.3

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

$\frac{3 \sqrt{5}}{5^{\frac{2}{2}}} \cdot \sqrt{0.2}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{5}}{5^{\frac{2}{2}}} \cdot \sqrt{0.2}$

Step 5.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{5}}{5^{1}} \cdot \sqrt{0.2}$

Step 5.6.5

Evaluate the exponent.

$\frac{3 \sqrt{5}}{5} \cdot \sqrt{0.2}$

Step 6

Multiply $\frac{3 \sqrt{5}}{5} \sqrt{0.2}$ .

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Step 6.1

Combine $\frac{3 \sqrt{5}}{5}$ and $\sqrt{0.2}$ .

$\frac{3 \sqrt{5} \sqrt{0.2}}{5}$

Step 6.2

Combine using the product rule for radicals.

$\frac{3 \sqrt{0.2 \cdot 5}}{5}$

Step 6.3

Multiply $0.2$ by $5$ .

$\frac{3 \sqrt{1}}{5}$

Step 7

Any root of $1$ is $1$ .

$\frac{3 \cdot 1}{5}$

Step 8

Multiply $3$ by $1$ .

$\frac{3}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{5}$

Decimal Form:

$0.6$