Simplify. $\newline$ $(\sqrt{5}-\sqrt{2})^{2}$

Full solution

Q. Simplify.

\newline

(\sqrt{5}-\sqrt{2})^{2}

Square the expression: We need to square the expression $\sqrt{5} - \sqrt{2}$ . Squaring a binomial means multiplying the binomial by itself. $\left(\sqrt{5} - \sqrt{2}\right)^2 = \left(\sqrt{5} - \sqrt{2}\right) \times \left(\sqrt{5} - \sqrt{2}\right)$
Apply distributive property: Now we will apply the distributive property (also known as the FOIL method for binomials) to multiply the two binomials. $\newline$ $(\sqrt{5} - \sqrt{2}) \times (\sqrt{5} - \sqrt{2}) = \sqrt{5} \times \sqrt{5} - \sqrt{5} \times \sqrt{2} - \sqrt{2} \times \sqrt{5} + \sqrt{2} \times \sqrt{2}$
Simplify terms: Next, we simplify each term. The square root of a number, squared, is just the number itself. So, $\sqrt{5} \times \sqrt{5} = 5$ and $\sqrt{2} \times \sqrt{2} = 2$ . The middle terms are like terms and can be combined. $5 - \sqrt{5} \times \sqrt{2} - \sqrt{2} \times \sqrt{5} + 2$
Combine like terms: Since $\sqrt{5} \times \sqrt{2}$ is the same as $\sqrt{5 \times 2}$ , we can simplify the middle terms. $\newline$ $5 - 2 \times \sqrt{5 \times 2} + 2$
Final result: Now we combine the like terms, which are the constants $5$ and $2$ . $5 + 2 - 2 \times \sqrt{10}$
Final result: Now we combine the like terms, which are the constants $5$ and $2$ . $5 + 2 - 2 \times \sqrt{10}$ Adding the constants together gives us: $7 - 2 \times \sqrt{10}$