To simplify the expression $\frac{\sqrt{20}}{\sqrt{2}}$√20√2 we can find a useful result from the exponent laws.
We know from the exponent laws that $\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$anbn=(ab)n. And we know that the index for a square root is $\frac{1}{2}$12. So if we substitute in $n=\frac{1}{2}$n=12 we get
$\frac{a^{\frac{1}{2}}}{b^{\frac{1}{2}}}=\left(\frac{a}{b}\right)^{\frac{1}{2}}$a12b12=(ab)12
Or written in radical form,
$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$√a√b=√ab
We can use this result to simplify our expression
$\frac{\sqrt{20}}{\sqrt{2}}$√20√2 | $=$= | $\sqrt{\frac{20}{2}}$√202 | (Simplifying into one radical) |
$=$= | $\sqrt{10}$√10 | (Simplifying the division under the radical) |
This can also be extended if we used division as notation in the form
$\sqrt{20}\div\sqrt{2}$√20÷√2 | $=$= | $\sqrt{20\div2}$√20÷2 | (Simplifying into one radical) |
$=$= | $\sqrt{10}$√10 | (Simplifying the division under the radical) |
Simplify the expression $\frac{4\sqrt{18}}{2\sqrt{3}}$4√182√3.
Think: For convenience we divide the radical parts and the non-radical parts separately. If we use the rule $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$√a√b=√ab, we can then simplify the fraction under the square root symbol.
Do:
$\frac{4\sqrt{18}}{2\sqrt{3}}$4√182√3 | $=$= | $\frac{4}{2}\frac{\sqrt{18}}{\sqrt{3}}$42√18√3 | (Separating division into radical and non-radical parts) |
$=$= | $\frac{4}{2}\sqrt{\frac{18}{3}}$42√183 | (Simplifying surds into one radical) | |
$=$= | $2\sqrt{6}$2√6 | (Simplifying the two divisions) |
Going back to the exponent law $\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$anbn=(ab)n, we can see that this will work not just for $n=\frac{1}{2}$n=12 but also $n=\frac{1}{3},\frac{1}{4},\frac{1}{5}...$n=13,14,15... and so on. That is, this will also work with cube roots, fourth roots, etc, as long as both roots have the same value of $n$n - that is, the same degree.
$\frac{\sqrt[k]{a}}{\sqrt[k]{b}}=\sqrt[k]{\frac{a}{b}}$k√ak√b=k√ab
Simplify the expression $\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$3√323√4.
Think: Using the rule that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}$3√a3√b=3√ab we can simplify the expression.
Do:
$\frac{\sqrt[3]{32}}{\sqrt[3]{4}}$3√323√4 | $=$= | $\sqrt[3]{\frac{32}{4}}$3√324 | (Simplifying into one radical) |
$=$= | $\sqrt[3]{8}$3√8 | (Simplifying division under the radical) | |
$=$= | $2$2 | (Simplifying the cube root expression) |
Simplify the expression $\frac{\sqrt{143}}{\sqrt{11}}$√143√11.
Simplify the expression $\frac{\sqrt[3]{40}}{\sqrt[3]{5}}$3√403√5.
Simplify the expression $\frac{21\sqrt{80}}{3\sqrt{5}}$21√803√5.