Simplify 4/( square root of 2)+2/( square root of 3)
The given problem is a mathematical expression that requires simplification. It involves two fractions, each with a numerator and a denominator. The numerators in both fractions are integers and the denominators are square roots of integers. The task is to combine these fractions into a single expression, ideally in its simplest form, which would typically involve rationalizing the denominators (eliminating the square roots from the denominators by appropriate multiplication). To fully simplify the expression, you may also need to find a common denominator for the fractions and combine like terms if possible.
$\frac{4}{\sqrt{2}} + \frac{2}{\sqrt{3}}$
Answer
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: $$3.98312766 \ldots$$
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.
