Simplify 4/( square root of 2)+2/( square root of 3)

Solution:

Step:1

Break down each term for simplification.

Step:1.1

Rationalize the denominator of $\frac{4}{\sqrt{2}}$ by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$.

$$\frac{4}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}+\frac{2}{\sqrt{3}}$$

Step:1.2

Simplify the resulting denominators.

Step:1.2.1

Rationalize $\frac{4}{\sqrt{2}}$ as follows:

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

Step:1.2.2

Express $\sqrt{2}$ as a power of $1$.

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

Step:1.2.3

Repeat the expression of $\sqrt{2}$ as a power of $1$.

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

Step:1.2.4

Apply the exponent combination rule.

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

Step:1.2.5

Sum the exponents $1$ and $1$.

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

Step:1.2.6

Convert $(\sqrt{2}{)}^2$ to $2$.

Step:1.2.6.1

Rewrite $\sqrt{2}$ using fractional exponents.

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

Step:1.2.6.2

Apply the exponent multiplication rule.

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

Step:1.2.6.3

Simplify the exponent $\frac{1}{2}\cdot 2$.

$$\frac{4\sqrt{2}}{2^{\frac{2}{2}}}+\frac{2}{\sqrt{3}}$$

Step:1.2.6.4

Simplify the exponent to $1$.

$$\frac{4\sqrt{2}}{2^1}+\frac{2}{\sqrt{3}}$$

Step:1.2.6.5

Evaluate the power of $2$.

$$\frac{4\sqrt{2}}{2}+\frac{2}{\sqrt{3}}$$

Step:1.3

Reduce the fraction by the common factor of $4$ and $2$.

Step:1.3.1

Extract $2$ from $4\sqrt{2}$.

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

Step:1.3.2

Eliminate the common factors.

Step:1.3.2.1

Factor out $2$ from $2$.

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

Step:1.3.2.2

Cancel out the common factor.

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

Step:1.3.2.3

Rewrite the simplified expression.

$$\frac{2\sqrt{2}}{1}+\frac{2}{\sqrt{3}}$$

Step:1.3.2.4

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

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

Step:1.4

Rationalize the denominator of $\frac{2}{\sqrt{3}}$ by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$.

$$2\sqrt{2}+\frac{2}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}$$

Step:1.5

Simplify the resulting denominators.

Step:1.5.1

Rationalize $\frac{2}{\sqrt{3}}$ as follows:

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

Step:1.5.2

Express $\sqrt{3}$ as a power of $1$.

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

Step:1.5.3

Repeat the expression of $\sqrt{3}$ as a power of $1$.

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

Step:1.5.4

Apply the exponent combination rule.

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

Step:1.5.5

Sum the exponents $1$ and $1$.

$$2\sqrt{2}+\frac{2\sqrt{3}}{(\sqrt{3}{)}^2}$$

Step:1.5.6

Convert $(\sqrt{3}{)}^2$ to $3$.

Step:1.5.6.1

Rewrite $\sqrt{3}$ using fractional exponents.

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

Step:1.5.6.2

Apply the exponent multiplication rule.

$$2\sqrt{2}+\frac{2\sqrt{3}}{3^{\frac{1}{2}\cdot 2}}$$

Step:1.5.6.3

Simplify the exponent $\frac{1}{2}\cdot 2$.

$$2\sqrt{2}+\frac{2\sqrt{3}}{3^{\frac{2}{2}}}$$

Step:1.5.6.4

Simplify the exponent to $1$.

$$2\sqrt{2}+\frac{2\sqrt{3}}{3^1}$$

Step:1.5.6.5

Evaluate the power of $3$.

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

Step:2

To express $2\sqrt{2}$ with a common denominator, multiply by $\frac{3}{3}$.

$$2\sqrt{2}\cdot \frac{3}{3}+\frac{2\sqrt{3}}{3}$$

Step:3

Combine $2\sqrt{2}$ with $\frac{3}{3}$.

$$\frac{2\sqrt{2}\cdot 3}{3}+\frac{2\sqrt{3}}{3}$$

Step:4

Simplify the combined expression.

Step:4.1

Merge the numerators over the common denominator.

$$\frac{2\sqrt{2}\cdot 3+2\sqrt{3}}{3}$$

Step:4.2

Multiply $2$ by $3$.

$$\frac{6\sqrt{2}+2\sqrt{3}}{3}$$

Step:5

The final result can be presented in different forms.

Exact Form: $$\frac{6\sqrt{2}+2\sqrt{3}}{3}$$ Decimal Form: $$3.98312766\dots$$

Knowledge Notes:

The problem involves simplifying an expression with square roots in the denominators. The process includes rationalizing the denominators, which means removing the square roots from the denominators by multiplying by a form of one that contains the square root, thus creating a rational number in the denominator. This is a common technique in algebra.

Key knowledge points include:

• Rationalizing the denominator
• Exponent rules, such as $a^m\cdot a^n=a^{m+n}$ and $(a^m{)}^n=a^{m\cdot n}$
• Simplifying expressions by combining like terms
• Understanding that $\sqrt[n]{a^x}=a^{\frac{x}{n}}$
• Working with common denominators to combine fractions

These concepts are foundational in algebra and are used to simplify expressions, solve equations, and perform operations with radicals and exponents.