Solve mu multiply the ( 6+ sqrt 3 sqrt 8 6- sqrt 3 )( 6- sqrt 3 sqrt 8 + sqrt 3 rational and irratio 8 mumber | 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

$$( \frac { 6 + \sqrt { 3 } } { \sqrt { 86 } - \sqrt { 3 } } ) ( \frac { 6 - \sqrt { 3 } } { \sqrt { 8 } + \sqrt { 3 } }$$

Evaluate

$\frac{66\sqrt{6}+132\sqrt{43}-33\sqrt{258}-99}{415}\approx 0.959493303$

Solution Steps

Rationalize the denominator of $\frac{6+\sqrt{3}}{\sqrt{86}-\sqrt{3}}$ by multiplying numerator and denominator by $\sqrt{86}+\sqrt{3}$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{\left(\sqrt{86}-\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}\times \frac{6-\sqrt{3}}{\sqrt{8}+\sqrt{3}}$$

Consider $\left(\sqrt{86}-\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{\left(\sqrt{86}\right)^{2}-\left(\sqrt{3}\right)^{2}}\times \frac{6-\sqrt{3}}{\sqrt{8}+\sqrt{3}}$$

Square $\sqrt{86}$. Square $\sqrt{3}$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{86-3}\times \frac{6-\sqrt{3}}{\sqrt{8}+\sqrt{3}}$$

Subtract $3$ from $86$ to get $83$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{6-\sqrt{3}}{\sqrt{8}+\sqrt{3}}$$

Factor $8=2^{2}\times 2$. Rewrite the square root of the product $\sqrt{2^{2}\times 2}$ as the product of square roots $\sqrt{2^{2}}\sqrt{2}$. Take the square root of $2^{2}$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{6-\sqrt{3}}{2\sqrt{2}+\sqrt{3}}$$

Rationalize the denominator of $\frac{6-\sqrt{3}}{2\sqrt{2}+\sqrt{3}}$ by multiplying numerator and denominator by $2\sqrt{2}-\sqrt{3}$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{\left(2\sqrt{2}+\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}$$

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

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{2^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Calculate $2$ to the power of $2$ and get $4$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

The square of $\sqrt{2}$ is $2$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{4\times 2-\left(\sqrt{3}\right)^{2}}$$

Multiply $4$ and $2$ to get $8$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{8-\left(\sqrt{3}\right)^{2}}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{8-3}$$

Subtract $3$ from $8$ to get $5$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}\times \frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{5}$$

Multiply $\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)}{83}$ times $\frac{\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{83\times 5}$$

Multiply $83$ and $5$ to get $415$.

$$\frac{\left(6+\sqrt{3}\right)\left(\sqrt{86}+\sqrt{3}\right)\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Apply the distributive property by multiplying each term of $6+\sqrt{3}$ by each term of $\sqrt{86}+\sqrt{3}$.

$$\frac{\left(6\sqrt{86}+6\sqrt{3}+\sqrt{3}\sqrt{86}+\left(\sqrt{3}\right)^{2}\right)\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

To multiply $\sqrt{3}$ and $\sqrt{86}$, multiply the numbers under the square root.

$$\frac{\left(6\sqrt{86}+6\sqrt{3}+\sqrt{258}+\left(\sqrt{3}\right)^{2}\right)\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{\left(6\sqrt{86}+6\sqrt{3}+\sqrt{258}+3\right)\left(6-\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Apply the distributive property by multiplying each term of $6\sqrt{86}+6\sqrt{3}+\sqrt{258}+3$ by each term of $6-\sqrt{3}$.

$$\frac{\left(36\sqrt{86}-6\sqrt{3}\sqrt{86}+36\sqrt{3}-6\left(\sqrt{3}\right)^{2}+6\sqrt{258}-\sqrt{258}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

To multiply $\sqrt{3}$ and $\sqrt{86}$, multiply the numbers under the square root.

$$\frac{\left(36\sqrt{86}-6\sqrt{258}+36\sqrt{3}-6\left(\sqrt{3}\right)^{2}+6\sqrt{258}-\sqrt{258}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{\left(36\sqrt{86}-6\sqrt{258}+36\sqrt{3}-6\times 3+6\sqrt{258}-\sqrt{258}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Multiply $-6$ and $3$ to get $-18$.

$$\frac{\left(36\sqrt{86}-6\sqrt{258}+36\sqrt{3}-18+6\sqrt{258}-\sqrt{258}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Combine $-6\sqrt{258}$ and $6\sqrt{258}$ to get $0$.

$$\frac{\left(36\sqrt{86}+36\sqrt{3}-18-\sqrt{258}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Factor $258=3\times 86$. Rewrite the square root of the product $\sqrt{3\times 86}$ as the product of square roots $\sqrt{3}\sqrt{86}$.

$$\frac{\left(36\sqrt{86}+36\sqrt{3}-18-\sqrt{3}\sqrt{86}\sqrt{3}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Multiply $\sqrt{3}$ and $\sqrt{3}$ to get $3$.

$$\frac{\left(36\sqrt{86}+36\sqrt{3}-18-3\sqrt{86}+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Combine $36\sqrt{86}$ and $-3\sqrt{86}$ to get $33\sqrt{86}$.

$$\frac{\left(33\sqrt{86}+36\sqrt{3}-18+18-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Add $-18$ and $18$ to get $0$.

$$\frac{\left(33\sqrt{86}+36\sqrt{3}-3\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Combine $36\sqrt{3}$ and $-3\sqrt{3}$ to get $33\sqrt{3}$.

$$\frac{\left(33\sqrt{86}+33\sqrt{3}\right)\left(2\sqrt{2}-\sqrt{3}\right)}{415}$$

Apply the distributive property by multiplying each term of $33\sqrt{86}+33\sqrt{3}$ by each term of $2\sqrt{2}-\sqrt{3}$.

$$\frac{66\sqrt{86}\sqrt{2}-33\sqrt{3}\sqrt{86}+66\sqrt{3}\sqrt{2}-33\left(\sqrt{3}\right)^{2}}{415}$$

Factor $86=2\times 43$. Rewrite the square root of the product $\sqrt{2\times 43}$ as the product of square roots $\sqrt{2}\sqrt{43}$.

$$\frac{66\sqrt{2}\sqrt{43}\sqrt{2}-33\sqrt{3}\sqrt{86}+66\sqrt{3}\sqrt{2}-33\left(\sqrt{3}\right)^{2}}{415}$$

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

$$\frac{66\times 2\sqrt{43}-33\sqrt{3}\sqrt{86}+66\sqrt{3}\sqrt{2}-33\left(\sqrt{3}\right)^{2}}{415}$$

Multiply $66$ and $2$ to get $132$.

$$\frac{132\sqrt{43}-33\sqrt{3}\sqrt{86}+66\sqrt{3}\sqrt{2}-33\left(\sqrt{3}\right)^{2}}{415}$$

To multiply $\sqrt{3}$ and $\sqrt{86}$, multiply the numbers under the square root.

$$\frac{132\sqrt{43}-33\sqrt{258}+66\sqrt{3}\sqrt{2}-33\left(\sqrt{3}\right)^{2}}{415}$$

To multiply $\sqrt{3}$ and $\sqrt{2}$, multiply the numbers under the square root.

$$\frac{132\sqrt{43}-33\sqrt{258}+66\sqrt{6}-33\left(\sqrt{3}\right)^{2}}{415}$$

The square of $\sqrt{3}$ is $3$.

$$\frac{132\sqrt{43}-33\sqrt{258}+66\sqrt{6}-33\times 3}{415}$$

Multiply $-33$ and $3$ to get $-99$.

$$\frac{132\sqrt{43}-33\sqrt{258}+66\sqrt{6}-99}{415}$$