Solve 1/(sqrt(2) + 3) - 3/(sqrt(2) - 1) | 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

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

Evaluate

$\frac{-22\sqrt{2}-18}{7}\approx -7.016099767$

Short Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{2}+3}$ by multiplying numerator and denominator by $\sqrt{2}-3$.

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

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

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

Square $\sqrt{2}$. Square $3$.

$$\frac{\sqrt{2}-3}{2-9}-\frac{3}{\sqrt{2}-1}$$

Subtract $9$ from $2$ to get $-7$.

$$\frac{\sqrt{2}-3}{-7}-\frac{3}{\sqrt{2}-1}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{2}+3}{7}-\frac{3}{\sqrt{2}-1}$$

Rationalize the denominator of $\frac{3}{\sqrt{2}-1}$ by multiplying numerator and denominator by $\sqrt{2}+1$.

$$\frac{-\sqrt{2}+3}{7}-\frac{3\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}$$

Consider $\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{-\sqrt{2}+3}{7}-\frac{3\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Square $\sqrt{2}$. Square $1$.

$$\frac{-\sqrt{2}+3}{7}-\frac{3\left(\sqrt{2}+1\right)}{2-1}$$

Subtract $1$ from $2$ to get $1$.

$$\frac{-\sqrt{2}+3}{7}-\frac{3\left(\sqrt{2}+1\right)}{1}$$

Anything divided by one gives itself.

$$\frac{-\sqrt{2}+3}{7}-3\left(\sqrt{2}+1\right)$$

Use the distributive property to multiply $3$ by $\sqrt{2}+1$.

$$\frac{-\sqrt{2}+3}{7}-\left(3\sqrt{2}+3\right)$$

To find the opposite of $3\sqrt{2}+3$, find the opposite of each term.

$$\frac{-\sqrt{2}+3}{7}-3\sqrt{2}-3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $-3\sqrt{2}-3$ times $\frac{7}{7}$.

$$\frac{-\sqrt{2}+3}{7}+\frac{7\left(-3\sqrt{2}-3\right)}{7}$$

Since $\frac{-\sqrt{2}+3}{7}$ and $\frac{7\left(-3\sqrt{2}-3\right)}{7}$ have the same denominator, add them by adding their numerators.

$$\frac{-\sqrt{2}+3+7\left(-3\sqrt{2}-3\right)}{7}$$

Do the multiplications in $-\sqrt{2}+3+7\left(-3\sqrt{2}-3\right)$.

$$\frac{-\sqrt{2}+3-21\sqrt{2}-21}{7}$$

Do the calculations in $-\sqrt{2}+3-21\sqrt{2}-21$.

$$\frac{-22\sqrt{2}-18}{7}$$