Solve 1/(sqrt(5) + 1); =,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

$$\frac{1}{\sqrt{5}+1}; =,1$$

Evaluate

$\frac{\sqrt{5}-1}{4}\approx 0.309016994$

Solution Steps

Rationalize the denominator of $\frac{1}{\sqrt{5}+1}$ by multiplying numerator and denominator by $\sqrt{5}-1$.

$$\frac{\sqrt{5}-1}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)}$$

Consider $\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\sqrt{5}-1}{\left(\sqrt{5}\right)^{2}-1^{2}}$$

Square $\sqrt{5}$. Square $1$.

$$\frac{\sqrt{5}-1}{5-1}$$

Subtract $1$ from $5$ to get $4$.

$$\frac{\sqrt{5}-1}{4}$$