Zero Exponent, Negative Exponents, and Root Forms

Zero Exponent

For every $a$ real number applies:

$\boxed{{a^0} = 1}$

Example

1. ${3^0} = 1$
2. ${\left( { - 4} \right)^0} = 1$
3. ${\left( {\frac{1}{2}} \right)^0} = 1$
4. ${y^0} = 1$
5. ${\left( {pq} \right)^0} = 1$

Negative Exponents

For every $a$ real number applies:

$\boxed{{a^{ - m}} = \frac{1}{{{a^m}}}}$

Example

1. ${2^{ - 3}} = \frac{1}{{{2^3}}} = \frac{1}{8}$
2. ${3^{ - 2}} = \frac{1}{{{3^2}}} = \frac{1}{9}$
3. ${\left( { - 2} \right)^{ - 4}} = \frac{1}{{{{\left( { - 2} \right)}^4}}} = \frac{1}{{16}}$
4. ${4^{ - 1}} = \frac{1}{{{4^1}}} = \frac{1}{4}$
5. ${2^{ - 2}} + {3^{ - 1}} = \frac{1}{{{2^2}}} + \frac{1}{{{3^1}}} = \frac{1}{4} + \frac{1}{3} = \frac{3}{{12}} + \frac{4}{{12}} = \frac{7}{{12}}$

Root Forms

The form of a root is the root of a rational number whose result is an irrational number. Examples of numbers that include the root form are $\sqrt 2 $, $\sqrt 3 $, $\sqrt 7 $, $\sqrt[3]{6}$, $\sqrt[4]{5}$ and many more. other. The thing to remember is that not all numbers in roots are in the form of roots. For example $\sqrt {16} $ is not the root form because $\sqrt {16} = 4$. $\sqrt {25} $ is not the root form because $\sqrt {25} = 5$.

Properties of Root Form

For $a$, $b$, $x$, and $y$ real numbers apply:

1. $\boxed{x\sqrt a \pm y\sqrt a = \left( {x \pm y} \right)\sqrt a} ;a \ge 0$
2. $\boxed{\sqrt a \times \sqrt b = \sqrt {a \times b}} ;a \ge 0;b \ge 0$
3. $\boxed{\frac{{\sqrt a }}{{\sqrt b }} = \sqrt {\frac{a}{b}}} ;a \ge 0;b \ge 0$

Example

1. $4\sqrt 2 + \sqrt 2 - 2\sqrt 2 = \left( {4 + 1 - 2} \right)\sqrt 2 = 3\sqrt 2 $
2. $\sqrt 3 \times \sqrt 5 = \sqrt {3 \times 5} = \sqrt {15} $
3. $2\sqrt 3 \times 3\sqrt 6 = \left( {2 \times 3} \right)\sqrt {3 \times 6} = 6\sqrt {18} $
4. $\frac{{\sqrt 6 }}{{\sqrt 2 }} = \sqrt {\frac{6}{2}} = \sqrt 3 $

Simplifying Root Forms

Simplifying the root form means changing the number in the root to be smaller/simple without changing its value.

Example

1. $\sqrt {48} = \sqrt {16 \times 3} = \sqrt {16} \times \sqrt 3 = 4 \times \sqrt 3 = 4\sqrt 3 $
2. $\sqrt {72} = \sqrt {36 \times 2} = \sqrt {36} \times \sqrt 2 = 6 \times \sqrt 2 = 6\sqrt 2 $
3. $\sqrt {242} = \sqrt {121 \times 2} = \sqrt {121} \times \sqrt 2 = 11 \times \sqrt 2 = 11\sqrt 2 $
4. $\sqrt {1.000} = \sqrt {100 \times 10} = \sqrt {100} \times \sqrt {10} = 10 \times \sqrt {10} = 10\sqrt {10} $

Rationalizing the Denominator

Rationalizing a denominator is changing a fraction whose denominator contains a root form to become a rational number.

Guidelines for Rationalizing Denominators

1. If the denominator is $b \sqrt a $, then multiply by $\frac{{\sqrt a }}{{\sqrt a }}$.
2. If the denominator is $\sqrt a + \sqrt b $, then multiply by $\frac{{\sqrt a - \sqrt b }}{{\sqrt a - \sqrt b }}$.
3. If the denominator is $\sqrt a - \sqrt b $, then multiply by $\frac{{\sqrt a + \sqrt b }}{{\sqrt a + \sqrt b }}$.
4. If the denominator is $a + \sqrt b $, then multiply by $\frac{{a - \sqrt b }}{{a - \sqrt b }}$.
5. If the denominator is $a - \sqrt b $, then multiply by $\frac{{a + \sqrt b }}{{a + \sqrt b }}$.

Example

1. $\frac{1}{{\sqrt 2 }} = \frac{1}{{\sqrt 2 }} \times \frac{{\sqrt 2 }}{{\sqrt 2 }} = \frac{{\sqrt 2 }}{2} = \frac{1}{2}\sqrt 2 $
2. $\frac{3}{{4\sqrt 2 }} = \frac{3}{{4\sqrt 2 }} \times \frac{{\sqrt 2 }}{{\sqrt 2 }} = \frac{{3\sqrt 2 }}{{4 \times 2}} = \frac{{3\sqrt 2 }}{8} = \frac{3}{8}\sqrt 2 $
3. $\frac{2}{{\sqrt 2 + \sqrt 3 }} = \frac{2}{{\sqrt 2 + \sqrt 3 }} \times \frac{{\sqrt 2 - \sqrt 3 }}{{\sqrt 2 - \sqrt 3 }} = \frac{{2\left( {\sqrt 2 - \sqrt 3 } \right)}}{{2 - 3}} = \frac{{2\left( {\sqrt 2 - \sqrt 3 } \right)}}{{ - 1}} = - 2\left( {\sqrt 2 - \sqrt 3 } \right) = - 2\sqrt 2 + 2\sqrt 3 $
4. $\frac{3}{{3 - \sqrt 5 }} = \frac{3}{{3 - \sqrt 5 }} \times \frac{{3 + \sqrt 5 }}{{3 + \sqrt 5 }} = \frac{{3\left( {3 + \sqrt 5 } \right)}}{{{3^2} - 5}} = \frac{{3\left( {3 + \sqrt 5 } \right)}}{{9 - 5}} = \frac{{3\left( {3 + \sqrt 5 } \right)}}{4} = \frac{3}{4}\left( {3 + \sqrt 5 } \right) = \frac{9}{4} + \frac{3}{4}\sqrt 5 $

Thus the discussion about zero exponent, negative exponents, and the form of roots. Hope it's useful