Solve m * sqrt(3) + n * sqrt(6) Express the form (3 + sqrt(2))/(sqrt(5) + sqrt(3)); (12 - sqrt(2))/(sqrt(5) + sqrt(3)) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { 3 + \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } } + \frac { 2 - \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 3 } }$$

Evaluate

$\frac{5\left(\sqrt{5}-\sqrt{3}\right)}{2}\approx 1.260042925$

Short Solution Steps

Rationalize the denominator of $\frac{3+\sqrt{2}}{\sqrt{5}+\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{5}-\sqrt{3}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}+\frac{2-\sqrt{2}}{\sqrt{5}+\sqrt{3}}$$

Consider $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{2-\sqrt{2}}{\sqrt{5}+\sqrt{3}}$$

Square $\sqrt{5}$. Square $\sqrt{3}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{5-3}+\frac{2-\sqrt{2}}{\sqrt{5}+\sqrt{3}}$$

Subtract $3$ from $5$ to get $2$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{2-\sqrt{2}}{\sqrt{5}+\sqrt{3}}$$

Rationalize the denominator of $\frac{2-\sqrt{2}}{\sqrt{5}+\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{5}-\sqrt{3}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$$

Consider $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square $\sqrt{5}$. Square $\sqrt{3}$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{5-3}$$

Subtract $3$ from $5$ to get $2$.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$$

Since $\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$ and $\frac{\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)+\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$$

Do the multiplications in $\left(3+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)+\left(2-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)$.

$$\frac{3\sqrt{5}-3\sqrt{3}+\sqrt{10}-\sqrt{6}+2\sqrt{5}-2\sqrt{3}-\sqrt{10}+\sqrt{6}}{2}$$

Do the calculations in $3\sqrt{5}-3\sqrt{3}+\sqrt{10}-\sqrt{6}+2\sqrt{5}-2\sqrt{3}-\sqrt{10}+\sqrt{6}$.

$$\frac{5\sqrt{5}-5\sqrt{3}}{2}$$

Factor

$\frac{5 {(\sqrt{5} - \sqrt{3})}}{2} = 1.2600429248272815$