Solve b . 1/(sqrt(2) + 1) + 1/(sqrt(2) - 1) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

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

Evaluate

$2\sqrt{2}\approx 2.828427125$

Solution Steps

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

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

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

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

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

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

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

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

Anything divided by one gives itself.

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

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

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

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

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

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

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

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

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

Anything divided by one gives itself.

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

Combine $\sqrt{2}$ and $\sqrt{2}$ to get $2\sqrt{2}$.

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

Add $-1$ and $1$ to get $0$.

$$2\sqrt{2}$$