Properties of Exponents (Rules)

Properties (rules) of exponents:

Zero power: $$${{a}}^{{0}}={1}$$$, $$${a}\ne{0}$$$
Zero base: $$${{0}}^{{a}}={0}$$$, $$${a}\ne{0}$$$
$$${{0}}^{{0}}$$$ is undefined
$$${{1}}^{{a}}={1}$$$
Negative exponent: $$${{a}}^{{-{b}}}=\frac{{1}}{{{a}}^{{b}}}$$$, $$${b}\ne{0}$$$
Nth root: $$${{a}}^{{\frac{{1}}{{n}}}}={\sqrt[{{n}}]{{{a}}}}$$$, $$${n}\ne{0}$$$
Addition of exponents: $$${{a}}^{{m}}\cdot{{a}}^{{n}}={{a}}^{{{m}+{n}}}$$$
Subtraction of exponents: $$$\frac{{{{a}}^{{m}}}}{{{{a}}^{{n}}}}={{a}}^{{{m}-{n}}}$$$, $$${a}\ne{0}$$$
Multiplication of exponents: $$${{\left({{a}}^{{m}}\right)}}^{{n}}={{a}}^{{{m}\cdot{n}}}={{\left({{a}}^{{n}}\right)}}^{{m}}$$$
Division of exponents: $$${\sqrt[{{n}}]{{{{a}}^{{m}}}}}={{a}}^{{\frac{{m}}{{n}}}}$$$, $$${n}\ne{0}$$$
$$${\sqrt[{{m}}]{{{{a}}^{{m}}}}}={a}$$$, if $$${m}$$$ is odd
$$${\sqrt[{{m}}]{{{{a}}^{{m}}}}}={\left|{a}\right|}$$$, if $$${m}$$$ is even
$$${\sqrt[{{n}}]{{{{a}}^{{m}}}}}={{\left({\sqrt[{{n}}]{{{a}}}}\right)}}^{{m}}$$$ (just pay attention to signs and check, whether number exists)
Power of a product: $$${{a}}^{{n}}\cdot{{b}}^{{n}}={{\left({a}{b}\right)}}^{{n}}$$$
Power of a quotient: $$$\frac{{{{a}}^{{n}}}}{{{{b}}^{{n}}}}={{\left(\frac{{a}}{{b}}\right)}}^{{n}}$$$, $$${b}\ne{0}$$$

We already covered all rules earlier, except last two.

To understand last two properties, consider the following example.

Example. Find $$${{2}}^{{3}}\cdot{{4}}^{{3}}$$$.

Let's rewrite numbers: $$${{\left({\color{red}{{{2}}}}\right)}}^{{3}}\cdot{{\left({\color{green}{{{4}}}}\right)}}^{{3}}={\color{red}{{{2}\cdot{2}\cdot{2}}}}\cdot{\color{green}{{{4}\cdot{4}\cdot{4}}}}$$$.

Now, regroup: $$${\color{red}{{{2}\cdot{2}\cdot{2}}}}\cdot{\left({\color{green}{{{4}\cdot{4}\cdot{4}}}}\right)}={\left({\color{red}{{{2}}}}\cdot{\color{green}{{{4}}}}\right)}\cdot{\left({\color{red}{{{2}}}}\cdot{\color{green}{{{4}}}}\right)}\cdot{\left({\color{red}{{{2}}}}\cdot{\color{green}{{{4}}}}\right)}={{\left({2}\cdot{4}\right)}}^{{3}}$$$.

Note, that on the last step, we wrapped the product, using exponent.

This property is valid for any exponent, so:

Power of a product: $$${\color{purple}{{{{a}}^{{n}}\cdot{{b}}^{{n}}={{\left({a}\cdot{b}\right)}}^{{n}}}}}$$$.

Similarly, it can be shown that $$$\frac{{{a}}^{{n}}}{{{b}}^{{n}}}={{\left(\frac{{a}}{{b}}\right)}}^{{n}}$$$.

Power of a quotient: $$${\color{purple}{{\frac{{{a}}^{{n}}}{{{b}}^{{n}}}={{\left(\frac{{a}}{{b}}\right)}}^{{n}}}}}$$$, $$${b}\ne{0}$$$.

We can combine above rules to simplify more complex examples.

Example 2. Find $$$\frac{{{6}}^{{4}}}{{{3}}^{{4}}}$$$.

Using power of a quotient rule, we can write, that $$$\frac{{{6}}^{{4}}}{{{3}}^{{4}}}={{\left(\frac{{6}}{{3}}\right)}}^{{4}}={{2}}^{{4}}={16}$$$.

Now, let's see how combination of rules works.

Example 3. Rewrite, using positive exponents: $$${\sqrt[{{5}}]{{{{3}}^{{2}}\cdot{{3}}^{{5}}}}}$$$.

We first apply rule for adding exponents: $$${\sqrt[{{5}}]{{{{3}}^{{2}}\cdot{{3}}^{{5}}}}}={\sqrt[{{5}}]{{{{3}}^{{{2}+{5}}}}}}={\sqrt[{{5}}]{{{{3}}^{{7}}}}}$$$.

Now, apply rule for dividing exponents: $$${\sqrt[{{5}}]{{{{3}}^{{7}}}}}={{3}}^{{\frac{{7}}{{5}}}}$$$.

So, $$${\sqrt[{{5}}]{{{{3}}^{{2}}\cdot{{3}}^{{5}}}}}={\sqrt[{{5}}]{{{{3}}^{{7}}}}}$$$.

Finally, let's see how to apply more than two rules.

Example 4. Rewrite, using positive exponents $$${{\left(\frac{{{\sqrt[{{5}}]{{{3}}}}\cdot{\sqrt[{{5}}]{{{4}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}$$$.

First, we rewrite using exponents: $$${{\left(\frac{{{\sqrt[{{5}}]{{{3}}}}\cdot{\sqrt[{{5}}]{{{4}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}={{\left(\frac{{{{3}}^{{\frac{{1}}{{5}}}}\cdot{{4}}^{{\frac{{1}}{{5}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}$$$.

Now, apply power of a product rule: $$${{\left(\frac{{{{3}}^{{\frac{{1}}{{5}}}}\cdot{{4}}^{{\frac{{1}}{{5}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}={{\left(\frac{{{{\left({3}\cdot{4}\right)}}^{{\frac{{1}}{{5}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}={{\left(\frac{{{{12}}^{{\frac{{1}}{{5}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}$$$.

Next, use rule for subtracting exponents: $$${{\left(\frac{{{{12}}^{{\frac{{1}}{{5}}}}}}{{{12}}^{{3}}}\right)}}^{{3}}={{\left({{12}}^{{\frac{{1}}{{5}}-{3}}}\right)}}^{{3}}={{\left({{12}}^{{-\frac{{14}}{{5}}}}\right)}}^{{3}}$$$.

Next, apply rule for multiplying exponents: $$${{\left({{12}}^{{-\frac{{14}}{{5}}}}\right)}}^{{3}}={{12}}^{{-\frac{{14}}{{5}}\cdot{3}}}={{12}}^{{-\frac{{42}}{{5}}}}$$$.

Finally, apply negative exponent rule: $$${{12}}^{{-\frac{{42}}{{5}}}}=\frac{{1}}{{{12}}^{{\frac{{42}}{{5}}}}}$$$.

Answer: $$$\frac{{1}}{{{12}}^{{\frac{{42}}{{5}}}}}$$$.

Now, practice a little.

Exercise 1. Rewrite, using positive exponents: $$${\sqrt[{{4}}]{{{2}\cdot{{2}}^{{5}}}}}$$$.

Answer: $$${{2}}^{{\frac{{3}}{{2}}}}$$$.

Exercise 2. Rewrite, using positive exponents: $$${{\left(\frac{{{2}}^{{3}}}{{{4}}^{{5}}}\cdot{\sqrt[{{5}}]{{{{5}}^{{7}}}}}\right)}}^{{0}}$$$.

Answer: $$${1}$$$. Hint: as long as base is non-zero, raising to zero power gives 1.

Exercise 3. Rewrite, using positive exponents: $$${\sqrt[{{5}}]{{{\sqrt[{{4}}]{{{{2}}^{{3}}\cdot{{6}}^{{3}}\cdot{{12}}^{{5}}}}}}}}$$$.

Answer: $$${{12}}^{{\frac{{2}}{{5}}}}$$$.

Exercise 4. Simplify: $$${{\left(\frac{{{{2}}^{{3}}}}{{{{6}}^{{3}}}}\cdot{{3}}^{{5}}\right)}}^{{7}}$$$.

Answer: $$${{3}}^{{14}}$$$.

Exercise 5. Rewrite, using positive exponents: $$${\sqrt[{{7}}]{{{{\left(\frac{{\sqrt[{{5}}]{{{{\left(-{2}\right)}}^{{7}}}}}}{{-{{2}}^{{3}}}}\cdot{{\left(-{2}\right)}}^{{7}}\right)}}^{{2}}}}}$$$.

Answer: $$${{2}}^{{\frac{{54}}{{35}}}}$$$. Hint. pay attention to signs: $$${{\left({{\left(-{2}\right)}}^{{\frac{{27}}{{5}}}}\right)}}^{{2}}={{2}}^{{\frac{{54}}{{5}}}}$$$. Minus vanishes, because we square.