Solve A = 1/(11 + 3sqrt(11)) + 1/(11 - 3sqrt(11)) | 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

$$A=\frac{1}{11+3\sqrt{11}}+\frac{1}{11-3\sqrt{11}}$$

Solve for A

$A=1$

Steps for Solving Linear Equation

Rationalize the denominator of $\frac{1}{11+3\sqrt{11}}$ by multiplying numerator and denominator by $11-3\sqrt{11}$.

$$A=\frac{11-3\sqrt{11}}{\left(11+3\sqrt{11}\right)\left(11-3\sqrt{11}\right)}+\frac{1}{11-3\sqrt{11}}$$

Consider $\left(11+3\sqrt{11}\right)\left(11-3\sqrt{11}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$A=\frac{11-3\sqrt{11}}{11^{2}-\left(3\sqrt{11}\right)^{2}}+\frac{1}{11-3\sqrt{11}}$$

Calculate $11$ to the power of $2$ and get $121$.

$$A=\frac{11-3\sqrt{11}}{121-\left(3\sqrt{11}\right)^{2}}+\frac{1}{11-3\sqrt{11}}$$

Expand $\left(3\sqrt{11}\right)^{2}$.

$$A=\frac{11-3\sqrt{11}}{121-3^{2}\left(\sqrt{11}\right)^{2}}+\frac{1}{11-3\sqrt{11}}$$

Calculate $3$ to the power of $2$ and get $9$.

$$A=\frac{11-3\sqrt{11}}{121-9\left(\sqrt{11}\right)^{2}}+\frac{1}{11-3\sqrt{11}}$$

The square of $\sqrt{11}$ is $11$.

$$A=\frac{11-3\sqrt{11}}{121-9\times 11}+\frac{1}{11-3\sqrt{11}}$$

Multiply $9$ and $11$ to get $99$.

$$A=\frac{11-3\sqrt{11}}{121-99}+\frac{1}{11-3\sqrt{11}}$$

Subtract $99$ from $121$ to get $22$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{1}{11-3\sqrt{11}}$$

Rationalize the denominator of $\frac{1}{11-3\sqrt{11}}$ by multiplying numerator and denominator by $11+3\sqrt{11}$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{\left(11-3\sqrt{11}\right)\left(11+3\sqrt{11}\right)}$$

Consider $\left(11-3\sqrt{11}\right)\left(11+3\sqrt{11}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{11^{2}-\left(-3\sqrt{11}\right)^{2}}$$

Calculate $11$ to the power of $2$ and get $121$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{121-\left(-3\sqrt{11}\right)^{2}}$$

Expand $\left(-3\sqrt{11}\right)^{2}$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{121-\left(-3\right)^{2}\left(\sqrt{11}\right)^{2}}$$

Calculate $-3$ to the power of $2$ and get $9$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{121-9\left(\sqrt{11}\right)^{2}}$$

The square of $\sqrt{11}$ is $11$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{121-9\times 11}$$

Multiply $9$ and $11$ to get $99$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{121-99}$$

Subtract $99$ from $121$ to get $22$.

$$A=\frac{11-3\sqrt{11}}{22}+\frac{11+3\sqrt{11}}{22}$$

Since $\frac{11-3\sqrt{11}}{22}$ and $\frac{11+3\sqrt{11}}{22}$ have the same denominator, add them by adding their numerators.

$$A=\frac{11-3\sqrt{11}+11+3\sqrt{11}}{22}$$

Do the calculations in $11-3\sqrt{11}+11+3\sqrt{11}$.

$$A=\frac{22}{22}$$

Divide $22$ by $22$ to get $1$.

$$A=1$$

Assign A

$A≔1$