Laws of Exponents

The expression:

$${\underbrace{(\sqrt{xy})\cdot(\sqrt{xy})\cdot(\sqrt{xy}) \ldots (\sqrt{xy})}_{(\sqrt{ 7 }+\sqrt{ 2 }) \text{ times}}\neq (\sqrt{xy})^{\sqrt{ 7 }+\sqrt{ 2 }}}$$

is not defined, since ${(\sqrt{ 7 }+\sqrt{ 2 })}$ is not a natural number.

$0^0$ is an indeterminate form.

$0^n$ is not defined for $n \in \mathbb{N}$.

$$\sqrt[2n+1]{(+)} = (+)$$

$$\sqrt[2n+1]{(-)} = (-)$$

$$\sqrt[2n]{(+)} = (+)$$

$$\sqrt[2n]{(-)} = \text{not defined in }\mathbb{R}$$

$$\boxed{P = a^n; \quad a \in \mathbb{R}, n \in \mathbb{N}, P \in \mathbb{R}}$$

Where:

• $a$: base
• $n$: natural exponent
• $P$: power

Properties

1. Product of like bases:

$$\boxed{x^m \cdot x^n = x^{m+n} \quad x \in \mathbb{R}, m, n \in \mathbb{N}}$$

2. Power of a power:

$$\boxed{(x^m)^n = x^{m \cdot n} \quad x \in \mathbb{R}, m, n \in \mathbb{N}}$$

$$\boxed{\left(((a^m)^n)^r\right)^s = a^{m \cdot n \cdot r \cdot s}}$$

3. Power of a product:

$$\boxed{(a \cdot b)^n = a^n \cdot b^n \quad a, b \in \mathbb{R}, n \in \mathbb{N}}$$

$$\boxed{(x^a \cdot y^b)^n = x^{a \cdot n} \cdot y^{b \cdot n}}$$

4. Division of like bases:

$$\boxed{\frac{a^m}{a^n} = a^{m-n} \quad m, n \in \mathbb{N}, m \geq n, a \in \mathbb{R} \setminus \{0\}}$$

5. Power of a quotient:

$$\boxed{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad n \in \mathbb{N}, b \in \mathbb{R} \setminus \{0\}}$$

6. Negative exponent of a fraction:

$$\boxed{\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n} \quad a, b \neq 0}$$

7. Successive Exponents

$$\boxed{x^{a^{b^c}} = x^{a^m}= x^n= z}$$

8. Absolute Value

$$\sqrt[2k]{x^{2k}} = |x| = \begin{cases} x & \text{if: } x > 0 \\ 0 & \text{if: } x = 0 \\ -x & \text{if: } x < 0 \end{cases}$$

$$\boxed{y = \sqrt[n]{x} \iff y^n = x \quad n \in \mathbb{N}, n \geq 2}$$

Properties

1. Root of a product:

$$\boxed{\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}}$$

If $n$ is even, then $a \geq 0$ and $b \geq 0$.

2. Root of a quotient:

$$\boxed{\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad b \neq 0}$$

If $n$ is even, then $a \geq 0$ and $b > 0$.

3. Root of a root:

$$\boxed{\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a} \quad m, n \in \mathbb{N}}$$

If $m \cdot n$ is even, then $a \geq 0$.

$$\boxed{\sqrt[m]{\sqrt[n]{\sqrt[s]{\sqrt[r]{a}}}} = \sqrt[m \cdot n \cdot s \cdot r]{a}}$$

4. Root of a power:

$$\boxed{\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}}$$

$$\boxed{\sqrt[ab]{x^{ac}} = \sqrt[b]{x^c} \quad \text{If } ab \text{ is even, then } x \in \mathbb{R}_0^+}$$

5. Nested Radicals

$$\sqrt[n]{x^{a}\cdot \sqrt[m]{x^{b}\cdot \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{(am + b)p+c}}$$

$$\sqrt[n]{x^{a}\div \sqrt[m]{x^{b}\div \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{(am - b)p+c}}$$

$$\sqrt[m]{x \cdot \sqrt[n]{y \cdot \sqrt[p]{z}}} = \sqrt[m]{x} \cdot \sqrt[m \cdot n]{y} \cdot \sqrt[n \cdot m \cdot p]{z}$$

$$\sqrt[m]{x^a \cdot \sqrt[n]{y^b \cdot \sqrt[p]{z^c}}} = \sqrt[m]{x^a} \cdot \sqrt[m \cdot n]{y^b} \cdot \sqrt[n \cdot m \cdot p]{z^c}$$

$$\sqrt[m]{x^{a}\div \sqrt[n]{y^{b}\div \sqrt[p]{z^{c}\div \sqrt[q]{w^{d}\div \sqrt[r]{v^{e}\div \sqrt[s]{u^f}}}}}} = \frac{\cfrac{a}{x^m} \cdot \cfrac{c}{z^{mnp}} \cdot \cfrac{e}{v^{mnpqr}}}{\cfrac{b}{y^{mn}} \cdot \cfrac{d}{w^{mnpq}} \cdot \cfrac{f}{u^{mnpqrs}}}$$

1. Equation with equal bases:

$$a^m = a^n \implies m = n; \quad a \gt 0 \land a \neq 1)$$

2. Equation with equal exponents:

$$a^m = b^m \implies a = b; \quad m \neq 0, a \gt 0 \land b \gt 0$$

3. Equation with different bases:

$$a^m = b^n \implies m = n =0$$

4. Explicit equation by reflection:

$$a^a = b^b \implies a = b; \quad a \neq 0 \land b \neq 0$$

5. Explicit equation by symmetry:

$$a^{a^{a+m}} = a^{a^{a+n}} \implies m = n$$