exponent

Exponentiation is an operation involving two numbers - the base and the exponent.

The division was a lie. Just like how subtraction was a lie. Because $2÷7=2\cdot 7^{-1}$.
The square root was also a lie. It’s just an exponent. Because $\sqrt[2]{7}=7^{\frac{1}{2}}$ and $\sqrt[3]{7}=7^{\frac{1}{3}}$.

Exponent identities.

$a^0=1$

$a^1=a$

$(a\cdot b{)}^c=a^c\cdot b^c$

$a^b\cdot a^c=a^{b+c}$

$(a^b{)}^c=a^{bc}$

$\frac{a^b}{a^c}=a^{b-c}$

$a^{-b}=\frac{1}{a^b}$

$a^{\frac{1}{b}}=\sqrt[b]{a}$

$\frac{a^c}{b^c}={\left(\frac{a}{b}\right)}^c$

$(a+b{)}^2=a^2+2ab+b^2$

$a^2-b^2=(a+b)(a-b)$

$(a+b{)}^3=a^3+3a{b}^2+3a^{2}b+b^3$

$a^3+b^3=(a+b)(a^2-ab+b^2)$

From the rules above, we can derive more.

${\left(\frac{a}{b}\right)}^c=a^c\cdot b^{-c}$

${\left(\frac{a}{b}\right)}^c=\frac{(a{)}^c}{(b{)}^c}=(a{)}^c\cdot {\left(\frac{1}{b}\right)}^c=a^c\cdot b^{-c}$

$a^2+ba={\left(a+\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2$

$a^2+ba+c=a{\left(b-(-\frac{b}{2a})\right)}^2+\left(c-\frac{b^2}{4a}\right)$