Algebra Examples

$i \sqrt{1 \frac{1}{2}} \cdot i \sqrt{\frac{4}{3}}$

Step 1

Convert $1 \frac{1}{2}$ to an improper fraction.

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Step 1.1

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

$i \sqrt{1 + \frac{1}{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 1.2

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

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Step 1.2.1

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

$i \sqrt{\frac{2}{2} + \frac{1}{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 1.2.2

Combine the numerators over the common denominator.

$i \sqrt{\frac{2 + 1}{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 1.2.3

Add $2$ and $1$ .

$i \sqrt{\frac{3}{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 2

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$i \frac{\sqrt{3}}{\sqrt{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 3

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$i (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) \cdot (i \sqrt{\frac{4}{3}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$i \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.2

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

$i \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.3

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

$i \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.4

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

$i \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.5

Add $1$ and $1$ .

$i \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6

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

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Step 4.6.1

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

$i \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6.2

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

$i \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6.3

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

$i \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$i \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6.4.2

Rewrite the expression.

$i \frac{\sqrt{3} \sqrt{2}}{2^{1}} \cdot (i \sqrt{\frac{4}{3}})$

Step 4.6.5

Evaluate the exponent.

$i \frac{\sqrt{3} \sqrt{2}}{2} \cdot (i \sqrt{\frac{4}{3}})$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$i \frac{\sqrt{3 \cdot 2}}{2} \cdot (i \sqrt{\frac{4}{3}})$

Step 5.2

Multiply $3$ by $2$ .

$i \frac{\sqrt{6}}{2} \cdot (i \sqrt{\frac{4}{3}})$

Step 6

Combine $i$ and $\frac{\sqrt{6}}{2}$ .

$\frac{i \sqrt{6}}{2} \cdot (i \sqrt{\frac{4}{3}})$

Step 7

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{\sqrt{4}}{\sqrt{3}})$

Step 8

Simplify the numerator.

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Step 8.1

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

$\frac{i \sqrt{6}}{2} \cdot (i \frac{\sqrt{2^{2}}}{\sqrt{3}})$

Step 8.2

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

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2}{\sqrt{3}})$

Step 9

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{i \sqrt{6}}{2} \cdot (i (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 10.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})$

Step 10.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})$

Step 10.4

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

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 10.5

Add $1$ and $1$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})$

Step 10.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 10.6.2

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

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 10.6.3

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

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 10.6.4.2

Rewrite the expression.

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{3^{1}})$

Step 10.6.5

Evaluate the exponent.

$\frac{i \sqrt{6}}{2} \cdot (i \frac{2 \sqrt{3}}{3})$

Step 11

Combine $i$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{i \sqrt{6}}{2} \cdot \frac{i (2 \sqrt{3})}{3}$

Step 12

Move $2$ to the left of $i$ .

$\frac{i \sqrt{6}}{2} \cdot \frac{2 i \sqrt{3}}{3}$

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Factor $2$ out of $2 i \sqrt{3}$ .

$\frac{i \sqrt{6}}{2} \cdot \frac{2 (i \sqrt{3})}{3}$

Step 13.2

Cancel the common factor.

$\frac{i \sqrt{6}}{2} \cdot \frac{2 (i \sqrt{3})}{3}$

Step 13.3

Rewrite the expression.

$i \sqrt{6} \cdot \frac{i \sqrt{3}}{3}$

Step 14

Combine $\frac{i \sqrt{3}}{3}$ and $i$ .

$\frac{i \sqrt{3} i}{3} \sqrt{6}$

Step 15

Raise $i$ to the power of $1$ .

$\frac{i^{1} i \sqrt{3}}{3} \sqrt{6}$

Step 16

Raise $i$ to the power of $1$ .

$\frac{i^{1} i^{1} \sqrt{3}}{3} \sqrt{6}$

Step 17

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

$\frac{i^{1 + 1} \sqrt{3}}{3} \sqrt{6}$

Step 18

Simplify terms.

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Step 18.1

Add $1$ and $1$ .

$\frac{i^{2} \sqrt{3}}{3} \sqrt{6}$

Step 18.2

Combine $\frac{i^{2} \sqrt{3}}{3}$ and $\sqrt{6}$ .

$\frac{i^{2} \sqrt{3} \sqrt{6}}{3}$

Step 18.3

Combine using the product rule for radicals.

$\frac{i^{2} \sqrt{6 \cdot 3}}{3}$

Step 18.4

Multiply $6$ by $3$ .

$\frac{i^{2} \sqrt{18}}{3}$

Step 19

Simplify the numerator.

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Step 19.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{- \sqrt{18}}{3}$

Step 19.2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 19.2.1

Factor $9$ out of $18$ .

$\frac{- \sqrt{9 (2)}}{3}$

Step 19.2.2

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

$\frac{- \sqrt{3^{2} \cdot 2}}{3}$

Step 19.3

Pull terms out from under the radical.

$\frac{- 1 \cdot 3 \sqrt{2}}{3}$

Step 19.4

Multiply $- 1$ by $3$ .

$\frac{- 3 \sqrt{2}}{3}$

Step 20

Cancel the common factor of $- 3$ and $3$ .

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Step 20.1

Factor $3$ out of $- 3 \sqrt{2}$ .

$\frac{3 (- \sqrt{2})}{3}$

Step 20.2

Cancel the common factors.

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Step 20.2.1

Factor $3$ out of $3$ .

$\frac{3 (- \sqrt{2})}{3 (1)}$

Step 20.2.2

Cancel the common factor.

$\frac{3 (- \sqrt{2})}{3 \cdot 1}$

Step 20.2.3

Rewrite the expression.

$\frac{- \sqrt{2}}{1}$

Step 20.2.4

Divide $- \sqrt{2}$ by $1$ .

$- \sqrt{2}$

Step 21

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$