Question

Surds and roots look at picture

Answer 1

$\huge \boxed{\mathfrak{Question} \downarrow}$

• Expand & simplify ⇨ $( \sqrt{10} - \sqrt{2} ) ^{2}$. Give your answer in the form $b - c \: \sqrt{5}$ where b & c are integers.

$\large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}$

$( \sqrt { 10 } - \sqrt { 2 } ) ^ { 2 }$

Use binomial theorem $\left(a-b\right)^{2}=a^{2}-2ab+b^{2}$ to expand $\left(\sqrt{10}-\sqrt{2}\right)^{2}$.

$\left(\sqrt{10}\right)^{2}-2\sqrt{10}\sqrt{2}+\left(\sqrt{2}\right)^{2}$

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

$10-2\sqrt{10}\sqrt{2}+\left(\sqrt{2}\right)^{2}$

Factor $10=2\times 5$. Rewrite the square root of the product $\sqrt{2\times 5}$ as the product of square roots $\sqrt{2}\sqrt{5}$.

$10-2\sqrt{2}\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}$

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

$10-2\times 2\sqrt{5}+\left(\sqrt{2}\right)^{2}$

Multiply -2 and 2 to get -4.

$10-4\sqrt{5}+\left(\sqrt{2}\right)^{2}$

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

$10-4\sqrt{5}+2$

Add 10 and 2 to get 12.

$\boxed{ \boxed{\bf\:12-4\sqrt{5} }}$

• Here, b & c are integers where $\boxed{ \sf \: b = 12 \: and \: c = 4}$

Answer 2

Answer:

$12 - 4 \sqrt{5}$

Step-by-step explanation:

${( \sqrt{10} - \sqrt{2} })^{2} \\ ( \sqrt{10} - \sqrt{2} )( \sqrt{10} - \sqrt{2} ) \\ 10 - 2 \sqrt{5} - 2 \sqrt{5} + 2 \\ 10 - 4 \sqrt{5} + 2 \\ 12 - 4 \sqrt{5}$

I hope I helped you^_^