Question

as you probably discovered, $sqrt{a + b}$ is not always equal to $sqrt{a}+sqrt{b}$ and $sqrt{a - b}$ may not equal $sqrt{a}-sqrt{b}$. we cant always combine square roots by addition or subtraction. sometimes we can do it by using the product and quotient rules to simplify the square roots. simplifying the square root of a whole number means finding an equivalent expression with the smallest possible number under the radical sign. this is called writing the number in simplest radical form. to write a square root in simplest radical form we first find the largest possible square factor of the number under the radical sign. then we use the product rule. look at this example. 4, 9 and 16 are squares less than 20. 4 is the only one which is a divisor of 20. $sqrt{20}=sqrt{4cdot5}=sqrt{4}sqrt{5}=2sqrt{5}$ rewrite each square root in simplest radical form. $sqrt{18}=sqrt{9}sqrt{2}=3sqrt{2}$ $sqrt{32}=$ $sqrt{500}=$ $sqrt{24}=$ $sqrt{50}=$ $sqrt{8}=$ if a number is large it is sometimes helpful to find its prime factors. $sqrt{80}=sqrt{2cdot2cdot2cdot2cdot5}=sqrt{4^{2}}sqrt{5}=4sqrt{5}$ $sqrt{162}=$ $sqrt{48}=$ $sqrt{108}=$ $sqrt{216}=$ $sqrt{405}=$ $sqrt{605}=$ $sqrt{96}=$

Explanation:

Step1: Find largest square factor of 32

$32 = 16\times2$, so $\sqrt{32}=\sqrt{16\times2}$

Step2: Apply product - rule

$\sqrt{16\times2}=\sqrt{16}\sqrt{2}=4\sqrt{2}$

Step3: For $\sqrt{24}$, find largest square factor

$24 = 4\times6$, so $\sqrt{24}=\sqrt{4\times6}$

Step4: Apply product - rule

$\sqrt{4\times6}=\sqrt{4}\sqrt{6}=2\sqrt{6}$

Step5: For $\sqrt{500}$, find largest square factor

$500 = 100\times5$, so $\sqrt{500}=\sqrt{100\times5}$

Step6: Apply product - rule

$\sqrt{100\times5}=\sqrt{100}\sqrt{5}=10\sqrt{5}$

Step7: For $\sqrt{50}$, find largest square factor

$50 = 25\times2$, so $\sqrt{50}=\sqrt{25\times2}$

Step8: Apply product - rule

$\sqrt{25\times2}=\sqrt{25}\sqrt{2}=5\sqrt{2}$

Step9: For $\sqrt{8}$, find largest square factor

$8 = 4\times2$, so $\sqrt{8}=\sqrt{4\times2}$

Step10: Apply product - rule

$\sqrt{4\times2}=\sqrt{4}\sqrt{2}=2\sqrt{2}$

Step11: For $\sqrt{162}$, find largest square factor

$162=81\times2$, so $\sqrt{162}=\sqrt{81\times2}$

Step12: Apply product - rule

$\sqrt{81\times2}=\sqrt{81}\sqrt{2}=9\sqrt{2}$

Step13: For $\sqrt{108}$, find largest square factor

$108 = 36\times3$, so $\sqrt{108}=\sqrt{36\times3}$

Step14: Apply product - rule

$\sqrt{36\times3}=\sqrt{36}\sqrt{3}=6\sqrt{3}$

Step15: For $\sqrt{405}$, find largest square factor

$405 = 81\times5$, so $\sqrt{405}=\sqrt{81\times5}$

Step16: Apply product - rule

$\sqrt{81\times5}=\sqrt{81}\sqrt{5}=9\sqrt{5}$

Step17: For $\sqrt{96}$, find largest square factor

$96 = 16\times6$, so $\sqrt{96}=\sqrt{16\times6}$

Step18: Apply product - rule

$\sqrt{16\times6}=\sqrt{16}\sqrt{6}=4\sqrt{6}$

Step19: For $\sqrt{605}$, find largest square factor

$605 = 121\times5$, so $\sqrt{605}=\sqrt{121\times5}$

Step20: Apply product - rule

$\sqrt{121\times5}=\sqrt{121}\sqrt{5}=11\sqrt{5}$

Step21: For $\sqrt{48}$, find largest square factor

$48 = 16\times3$, so $\sqrt{48}=\sqrt{16\times3}$

Step22: Apply product - rule

$\sqrt{16\times3}=\sqrt{16}\sqrt{3}=4\sqrt{3}$

Step23: For $\sqrt{216}$, find largest square factor

$216 = 36\times6$, so $\sqrt{216}=\sqrt{36\times6}$

Step24: Apply product - rule

$\sqrt{36\times6}=\sqrt{36}\sqrt{6}=6\sqrt{6}$

Answer:

$\sqrt{32}=4\sqrt{2}$
$\sqrt{24}=2\sqrt{6}$
$\sqrt{500}=10\sqrt{5}$
$\sqrt{50}=5\sqrt{2}$
$\sqrt{8}=2\sqrt{2}$
$\sqrt{162}=9\sqrt{2}$
$\sqrt{108}=6\sqrt{3}$
$\sqrt{405}=9\sqrt{5}$
$\sqrt{96}=4\sqrt{6}$
$\sqrt{605}=11\sqrt{5}$
$\sqrt{48}=4\sqrt{3}$
$\sqrt{216}=6\sqrt{6}$