Solve 4sqrt(7 1/2) - (2sqrt(10))/(2sqrt(3) - sqrt(10)) + 8 + 3sqrt(10) ni | 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

$$4 \sqrt { 7 \frac { 1 } { 2 } } - \frac { 2 \sqrt { 10 } } { 2 \sqrt { 3 } - \sqrt { 10 } } + 8 + 3 \sqrt { 10 } n i$$

Evaluate

$\left(\sqrt{10}+2\sqrt{3}\right)\left(3\sqrt{30}in-15in+\sqrt{10}-2\sqrt{3}\right)$

Solution Steps

Multiply $7$ and $2$ to get $14$.

$$4\sqrt{\frac{14+1}{2}}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

Add $14$ and $1$ to get $15$.

$$4\sqrt{\frac{15}{2}}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

Rewrite the square root of the division $\sqrt{\frac{15}{2}}$ as the division of square roots $\frac{\sqrt{15}}{\sqrt{2}}$.

$$4\times \frac{\sqrt{15}}{\sqrt{2}}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

Rationalize the denominator of $\frac{\sqrt{15}}{\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{2}$.

$$4\times \frac{\sqrt{15}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

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

$$4\times \frac{\sqrt{15}\sqrt{2}}{2}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

To multiply $\sqrt{15}$ and $\sqrt{2}$, multiply the numbers under the square root.

$$4\times \frac{\sqrt{30}}{2}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

Cancel out $2$, the greatest common factor in $4$ and $2$.

$$2\sqrt{30}-\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}+8+3i\sqrt{10}n$$

Rationalize the denominator of $\frac{2\sqrt{10}}{2\sqrt{3}-\sqrt{10}}$ by multiplying numerator and denominator by $2\sqrt{3}+\sqrt{10}$.

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{\left(2\sqrt{3}-\sqrt{10}\right)\left(2\sqrt{3}+\sqrt{10}\right)}+8+3i\sqrt{10}n$$

Consider $\left(2\sqrt{3}-\sqrt{10}\right)\left(2\sqrt{3}+\sqrt{10}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{10}\right)^{2}}+8+3i\sqrt{10}n$$

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

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{10}\right)^{2}}+8+3i\sqrt{10}n$$

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

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{4\left(\sqrt{3}\right)^{2}-\left(\sqrt{10}\right)^{2}}+8+3i\sqrt{10}n$$

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

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{4\times 3-\left(\sqrt{10}\right)^{2}}+8+3i\sqrt{10}n$$

Multiply $4$ and $3$ to get $12$.

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{12-\left(\sqrt{10}\right)^{2}}+8+3i\sqrt{10}n$$

The square of $\sqrt{10}$ is $10$.

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{12-10}+8+3i\sqrt{10}n$$

Subtract $10$ from $12$ to get $2$.

$$2\sqrt{30}-\frac{2\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)}{2}+8+3i\sqrt{10}n$$

Cancel out $2$ and $2$.

$$2\sqrt{30}-\sqrt{10}\left(2\sqrt{3}+\sqrt{10}\right)+8+3i\sqrt{10}n$$

Use the distributive property to multiply $\sqrt{10}$ by $2\sqrt{3}+\sqrt{10}$.

$$2\sqrt{30}-\left(2\sqrt{10}\sqrt{3}+\left(\sqrt{10}\right)^{2}\right)+8+3i\sqrt{10}n$$

To multiply $\sqrt{10}$ and $\sqrt{3}$, multiply the numbers under the square root.

$$2\sqrt{30}-\left(2\sqrt{30}+\left(\sqrt{10}\right)^{2}\right)+8+3i\sqrt{10}n$$

The square of $\sqrt{10}$ is $10$.

$$2\sqrt{30}-\left(2\sqrt{30}+10\right)+8+3i\sqrt{10}n$$

To find the opposite of $2\sqrt{30}+10$, find the opposite of each term.

$$2\sqrt{30}-2\sqrt{30}-10+8+3i\sqrt{10}n$$

Combine $2\sqrt{30}$ and $-2\sqrt{30}$ to get $0$.

$$-10+8+3i\sqrt{10}n$$

Add $-10$ and $8$ to get $-2$.

$$-2+3i\sqrt{10}n$$