How to use them
Each part has its solution folded away. Do the exercise in your notebook and, once you have it, click on the statement to unfold the solution and check it.
The properties of powers and radicals are in section 5.
Radicals
1
Complete the table
| Radical | $\sqrt[3]{2}$ | $\square$ | $\square$ | $\sqrt{x}$ | $\square$ | $\dfrac{1}{\sqrt[3]{3}}$ |
| Power | $\square$ | $25^{\frac{3}{4}}$ | $3^{-\frac{1}{5}}$ | $\square$ | $16^{\frac{3}{4}}$ | $\square$ |
See the solution
The exponent of the power is a fraction: the numerator is the exponent inside and the denominator is the index.
$$\sqrt[3]{2} = 2^{\frac{1}{3}} \qquad \sqrt[4]{25^{3}} = 25^{\frac{3}{4}} \qquad \sqrt[5]{3^{-1}} = \sqrt[5]{\tfrac{1}{3}} = 3^{-\frac{1}{5}}$$
$$\sqrt{x} = x^{\frac{1}{2}} \qquad \sqrt[4]{16^{3}} = 16^{\frac{3}{4}} = 8 \qquad \frac{1}{\sqrt[3]{3}} = \sqrt[3]{\tfrac{1}{3}} = \left(\tfrac{1}{3}\right)^{\frac{1}{3}}$$
2
Simplify by taking every possible factor out of the root
a) $\sqrt{8}$
$$\sqrt{8} = \sqrt{2^{3}} = 2\sqrt{2}$$
b) $\sqrt[3]{16}$
$$\sqrt[3]{16} = \sqrt[3]{2^{4}} = 2\sqrt[3]{2}$$
c) $\sqrt[5]{2^{11}}$
$$\sqrt[5]{2^{11}} = 2^{2}\sqrt[5]{2}$$
$11 = 2 \cdot 5 + 1$: two $2$s come out and one stays inside.
d) $\sqrt[10]{x^{14}}$
$$\sqrt[10]{x^{14}} = x\sqrt[10]{x^{4}} = x\sqrt[5]{x^{2}}$$
At the end the index is simplified: divide index and exponent by $2$.
e) $\sqrt[3]{x^{25}}$
$$\sqrt[3]{x^{25}} = x^{8}\sqrt[3]{x}$$
f) $\sqrt[3]{x^{15}}$
$$\sqrt[3]{x^{15}} = x^{5}$$
3
Simplify the following expressions
a) $\sqrt{a^{3} \cdot 27}$
$$\sqrt{a^{3} \cdot 27} = \sqrt{a^{2} \cdot a \cdot 3^{2} \cdot 3} = 3a\sqrt{3a}$$
b) $\sqrt[3]{648}$
$$\sqrt[3]{648} = \sqrt[3]{2^{3} \cdot 3^{4}} = 2 \cdot 3\sqrt[3]{3} = 6\sqrt[3]{3}$$
c) $\sqrt[3]{3^{6} \cdot 2^{11}}$
$$\sqrt[3]{3^{6} \cdot 2^{11}} = 3^{2} \cdot 2^{3}\sqrt[3]{2^{2}} = 72\sqrt[3]{4}$$
d) $\sqrt[5]{-64x^{6}}$
$$\sqrt[5]{-64x^{6}} = \sqrt[5]{-2^{6}x^{6}} = -2x\sqrt[5]{2x}$$
The index is odd, so the minus sign can come out of the root.
e) $\sqrt[3]{\dfrac{81}{x^{5}}}$
$$\sqrt[3]{\frac{81}{x^{5}}} = \sqrt[3]{\frac{3^{4}}{x^{5}}} = \frac{3}{x}\sqrt[3]{\frac{3}{x^{2}}}$$
f) $\sqrt[5]{\dfrac{1}{64}}$
$$\sqrt[5]{\frac{1}{64}} = \sqrt[5]{\frac{1}{2^{6}}} = \frac{1}{2}\sqrt[5]{\frac{1}{2}}$$
4
Reduce the following radicals to a common index
a) $\sqrt[6]{a^{3}},\ \sqrt[3]{a^{2}},\ \sqrt{a^{3}}$
Common index: $\text{lcm}(6,3,2) = 6$.
$$\sqrt[6]{a^{3}},\quad \sqrt[6]{a^{4}},\quad \sqrt[6]{a^{9}}$$
b) $\sqrt{a},\ \sqrt[3]{a},\ \sqrt[5]{a}$
Common index: $\text{lcm}(2,3,5) = 30$.
$$\sqrt[30]{a^{15}},\quad \sqrt[30]{a^{10}},\quad \sqrt[30]{a^{6}}$$
c) $\sqrt[5]{a},\ \sqrt{b^{5}},\ \sqrt{c^{3}}$
Common index: $\text{lcm}(5,2,2) = 10$.
$$\sqrt[10]{a^{2}},\quad \sqrt[10]{b^{25}},\quad \sqrt[10]{c^{15}}$$
d) $\sqrt{8},\ \sqrt[10]{2},\ \sqrt[5]{16}$
Common index: $\text{lcm}(2,10,5) = 10$.
$$\sqrt[10]{8^{5}} = \sqrt[10]{2^{15}},\quad \sqrt[10]{2},\quad \sqrt[10]{16^{2}} = \sqrt[10]{2^{8}}$$
Operations with radicals
1
Carry out the operations and simplify
a) $\sqrt[3]{3^{8}} \cdot \sqrt[3]{3^{-5}}$
$$\sqrt[3]{3^{8}} \cdot \sqrt[3]{3^{-5}} = \sqrt[3]{3^{8-5}} = \sqrt[3]{3^{3}} = 3$$
b) $\sqrt{6} \cdot \sqrt{2^{5}} \cdot \sqrt{24}$
$$\sqrt{2 \cdot 3 \cdot 2^{5} \cdot 2^{3} \cdot 3} = \sqrt{2^{9} \cdot 3^{2}} = 2^{4} \cdot 3\sqrt{2} = 48\sqrt{2}$$
c) $\sqrt[3]{x^{5}} \cdot \sqrt[3]{(2x)^{3}} \cdot \sqrt[3]{2x^{4}}$
$$\sqrt[3]{x^{5} \cdot 2^{3}x^{3} \cdot 2x^{4}} = \sqrt[3]{2^{4} \cdot x^{12}} = 2x^{4}\sqrt[3]{2}$$
d) $\dfrac{\sqrt[5]{64}}{\sqrt[5]{2^{-4}}}$
$$\sqrt[5]{\frac{2^{6}}{2^{-4}}} = \sqrt[5]{2^{10}} = 2^{2} = 4$$
e) $\dfrac{\sqrt{75}}{\sqrt{48}}$
$$\sqrt{\frac{75}{48}} = \sqrt{\frac{3 \cdot 5^{2}}{3 \cdot 2^{4}}} = \sqrt{\frac{5^{2}}{2^{4}}} = \frac{5}{2^{2}} = \frac{5}{4}$$
f) $\sqrt[3]{2^{5}} \cdot 2^{-\frac{2}{3}}$
With powers:
$$2^{\frac{5}{3}} \cdot 2^{-\frac{2}{3}} = 2^{\frac{5}{3} - \frac{2}{3}} = 2^{\frac{3}{3}} = 2$$
Or with radicals:
$$\sqrt[3]{2^{5}} \cdot \sqrt[3]{2^{-2}} = \sqrt[3]{2^{3}} = 2$$
2
Calculate and simplify
a) $\sqrt{8} + \sqrt{18} - \sqrt{32}$
$$\sqrt{2^{3}} + \sqrt{2 \cdot 3^{2}} - \sqrt{2^{5}} = 2\sqrt{2} + 3\sqrt{2} - 4\sqrt{2} = \sqrt{2}$$
b) $3\sqrt{75} - 5\sqrt{12} - \sqrt{3} + \sqrt{27}$
$$3\sqrt{3 \cdot 5^{2}} - 5\sqrt{2^{2} \cdot 3} - \sqrt{3} + \sqrt{3^{3}}$$
$$= 15\sqrt{3} - 10\sqrt{3} - \sqrt{3} + 3\sqrt{3} = 7\sqrt{3}$$
c) $3\sqrt{32x^{3}} + \sqrt{50x} - \sqrt{200x^{3}} - 2\sqrt{8x}$
$$3\sqrt{2^{5}x^{3}} + \sqrt{2 \cdot 5^{2}x} - \sqrt{2^{3} \cdot 5^{2}x^{3}} - 2\sqrt{2^{3}x}$$
$$= 12x\sqrt{2x} + 5\sqrt{2x} - 10x\sqrt{2x} - 4\sqrt{2x}$$
$$= 2x\sqrt{2x} + \sqrt{2x} = (2x + 1)\sqrt{2x}$$
Only like radicals can be added: same index and same radicand, here $\sqrt{2x}$.
d) $\sqrt[3]{\dfrac{16}{375}} + \sqrt{\dfrac{100x}{3}} + \sqrt[3]{\dfrac{128}{81}} + \sqrt{\dfrac{4x}{27}}$
$$\sqrt[3]{\frac{2^{4}}{3 \cdot 5^{3}}} + \sqrt{\frac{2^{2} \cdot 5^{2}\, x}{3}} + \sqrt[3]{\frac{2^{7}}{3^{4}}} + \sqrt{\frac{2^{2}x}{3^{3}}}$$
$$= \frac{2}{5}\sqrt[3]{\frac{2}{3}} + 10\sqrt{\frac{x}{3}} + \frac{4}{3}\sqrt[3]{\frac{2}{3}} + \frac{2}{3}\sqrt{\frac{x}{3}}$$
$$= \left(\frac{2}{5} + \frac{4}{3}\right)\sqrt[3]{\frac{2}{3}} + \left(10 + \frac{2}{3}\right)\sqrt{\frac{x}{3}} = \frac{26}{15}\sqrt[3]{\frac{2}{3}} + \frac{32}{3}\sqrt{\frac{x}{3}}$$