These examples use the product rule $\left(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\right)$ and the quotient rule $\left(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\right)$.
(Note: The provided image shows $\sqrt{32} = \sqrt{2\times 2\times 2\times 2\times 2} = 2$, which is mathematically incorrect. $\sqrt{32}$ simplifies to $4\sqrt{2}$ and is approximately $5.657$. The final line of calculation in the image is likely a typo and should be ignored, or it’s calculating $\sqrt[5]{32}=2$. Assuming it meant square roots, the correct answer is $\mathbf{4\sqrt{2}}$.)
(Note: The provided image shows $\sqrt{81} = \sqrt{9^2} = 3$. This is incorrect. $\sqrt{81}$ is $\mathbf{9}$. The image likely miswrote $9$ as $3$ or was aiming for $\sqrt[4]{81}=3$. Assuming a square root, the correct answer is $\mathbf{9}$.)
These examples apply standard algebraic formulas, especially the difference of squares $(a+b)(a-b) = a^2 – b^2$ and the square of a binomial $(a+b)^2 = a^2 + 2ab + b^2$.
(Note: The image shows an error in the multiplication and simplification, resulting in $3+\sqrt{3}$. The correct expansion has four separate terms and cannot be simplified further: $\mathbf{6 – 2\sqrt{5} + 3\sqrt{3} – \sqrt{15}}$)
(Note: The image shows $5-2\sqrt{6}$. The $\sqrt{10}$ part is correct in the original image’s last line of calculation: $5-2\sqrt{6}$. There is a slight error in the image calculation where $\sqrt{5\times 2}$ should be $\sqrt{10}$, not $\sqrt{6}$. Assuming the expression was $(\sqrt{3}-\sqrt{2})^2$ instead, the answer would be $3+2-2\sqrt{6}=5-2\sqrt{6}$. Based on the written expression $(\sqrt{5}-\sqrt{2})^2$, the correct final answer is $\mathbf{7 – 2\sqrt{10}}$)