Exponents

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Lesson Plan ▼

Lesson Plan

Exponentiation

Mathematical operation,
defines how many times to use the number in a multiplication

$$\bbox[8pt]{\Large{{\color{red}{a}}^{{\color{blue}{n}}} = \underbrace{{\color{red}{a}}\cdot {\color{red}{a}}\cdot {\color{red}{a}}\, {\color{gray}{\ldots}} \cdot {\color{red}{a}}}_{{\color{blue}{n}}}}};\quad {\color{blue}{n}} \in \textbf{N}$$

$a^5 = \underbrace{a\cdot a\cdot a \cdot a \cdot a}_{5}$

Notations

$$\bbox[8pt]{\Large{{\color{red}{\text{base}}}^{{\color{blue}{\text{exponent}}}}}}$$

ExpressionBaseExponentExponentiation
$2^3$${\color{red}{2}}$${\color{blue}{3}}$$2^3=2\cdot 2\cdot 2 = 8$
$3^2$${\color{red}{3}}$${\color{blue}{2}}$$3^2=3\cdot 3 = 9$
$5^2$${\color{red}{5}}$${\color{blue}{2}}$$5^2=5\cdot 5 = 25$
$4^3$${\color{red}{4}}$${\color{blue}{3}}$$4^3=4\cdot 4\cdot 4 = 64$
$8^1$${\color{red}{8}}$${\color{blue}{1}}$$8^1=8$

In words

ExpressionIs called
${\color{red}{a}}^{\color{blue}{2}}$${\color{red}{a}}$ to the power ${\color{blue}{2}}$
${\color{red}{a}}$ squared
${\color{red}{a}}$ to the second power
${\color{red}{a}}^{\color{blue}{3}}$${\color{red}{a}}$ to the power ${\color{blue}{3}}$
${\color{red}{a}}$ cubed
${\color{red}{a}}$ to the third power
${\color{red}{a}}^{\color{blue}{4}}$${\color{red}{a}}$ to the power ${\color{blue}{4}}$
${\color{red}{a}}$ to the 4th
${\color{red}{a}}$ to the fourth power

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

Exponential identities

$a^{\color{red}{2}}\cdot a^{\color{blue}{3}}={\color{red}{a}}\cdot{\color{red}{a}}\cdot{\color{blue}{a}}\cdot{\color{blue}{a}}\cdot{\color{blue}{a}}=a^{{\color{red}{2}}+{\color{blue}{3}}}=a^{5}$

$7^2\cdot 7^3 = (\underbrace{7\cdot 7}_{7^{2}}) \cdot (\underbrace{7\cdot 7 \cdot 7 }_{7^{3}}) = 7^{2+3}=7^{5}$

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

${\color{blue}{3}}^2\cdot {\color{blue}{3}}^5=$ $\,3^{2+5}=3^{7}$

$2{\color{red}{a}}^3\cdot {\color{red}{a}}^5=$$\,2a^{3+5}=2a^8$

$4{\color{red}{a}}^2\cdot {\color{red}{a}}^{3}b=$$\,4a^{2+3}b=4a^{5}b$

$5{\color{red}{a}}^{2}{\color{blue}{b}}^{3}\cdot {\color{red}{a}}^{4}{\color{blue}{b}}^{5}c^{3}=$$\,5a^{2+4}b^{3+5}c^3=5a^{6}b^{8}c^3$

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

${\color{red}{a}}^2{\color{blue}{b}}^3 \cdot(2{\color{red}{a}}+3{\color{red}{a}}^{5}{\color{blue}{b}}^{2})=$$\,2a^{2+1}b^{3}+3a^{2+5}b^{3+2}$

${\color{red}{a}}{\color{blue}{b}}^2 \cdot(ab^2+cb)=$$\,a^2b^4+ab^3c$

$a^2b^3c \cdot(abc^2-bc^2)=$$\,a^3b^4c^3-a^2b^4c^3$

$2ab\cdot(2abc+ab)=$$\,(2ab)^2c+2(ab)^2$

$(a^n)^m = a^{n\cdot m}$

Exponential identities

$\left(a^{\color{red}{2}}\right)^{\color{blue}{3}} = ({\color{red}{a}} \cdot {\color{red}{a}})^{\color{blue}{3}}=\underbrace{(a\cdot a)\cdot(a\cdot a)\cdot(a\cdot a)}_{\color{blue}{3}}=a^{{\color{red}{2}}\cdot {\color{blue}{3}}}$

$\left(5^3\right)^2 = \underbrace{(5\cdot 5\cdot 5)\cdot(5\cdot 5\cdot 5)}_{2}=$
$ = (\underbrace{5\cdot 5 \cdot 5\cdot 5\cdot 5 \cdot 5 }_{5^{6}}) = 5^{3\cdot 2} = 5^6$

$(a^n)^m = a^{n\cdot m}$

$\left(x^{\color{red}{2}}\right)^{\color{blue}{5}}=$ $\,x^{{\color{red}{2}}\cdot{\color{blue}{5}}}=x^{10}$

$\left(2^{\color{red}{3}}\right)^{\color{blue}{4}}=$$\,2^{{\color{red}{3}}\cdot{\color{blue}{4}}}=2^{12}$

$\left(\left(5+x\right)^{\color{red}{2}}\right)^{\color{blue}{3}}=$$\,\left(5+x\right)^{{\color{red}{2}}\cdot{\color{blue}{3}}}=\left(5+x\right)^{12}$

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

Exponential identities

$(a\cdot b)^2=({\color{red}{a}}\cdot{\color{blue}{b}})\cdot({\color{red}{a}}\cdot{\color{blue}{b}})={\color{red}{a}}\cdot {\color{red}{a}}\cdot {\color{blue}{b}} \cdot {\color{blue}{b}}=a^{2}\cdot b^2$

$(5\cdot 7)^3 = \underbrace{(5\cdot 7)\cdot(5\cdot 7)\cdot(5\cdot 7)}_{3}=$
$ = (\underbrace{5\cdot 5 \cdot 5 }_{5^{3}})\cdot (\underbrace{7\cdot 7 \cdot 7 }_{7^{3}}) = 5^3\cdot 7^3$

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

$\left(2x\right)^{\color{blue}{3}}=$ $\,2^{\color{blue}{3}}\cdot x^{\color{blue}{3}}=8x^3$

$\left(ab^2\right)^{\color{blue}{3}}=$$\,a^{\color{blue}{3}}\cdot b^{2\cdot {\color{blue}{3}}}=a^{3}b^{6}$

$\left(3x^2y^5\right)^{\color{blue}{2}}=$$\,3^{\color{blue}{2}}\cdot x^{2\cdot {\color{blue}{2}}}\cdot y^{5\cdot {\color{blue}{2}}}=9x^{4}y^{10}$

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

$\left({\color{red}{a}}\cdot ({\color{blue}{x+y}})\right)^2=$ $\,{\color{red}{a}}^{2}\cdot ({\color{blue}{x+y}})^{2}$

$\left({\color{red}{2xy^5}}\cdot \left({\color{blue}{3x^3y}}\right)^3\right)^2=$$\,\left({\color{red}{2xy^5}}\right)^{2}\cdot \left({\color{blue}{3x^3y}}\right)^{6}=$
$=2^{2}x^{2}y^{10}\cdot 3^{6}x^{18}y^{6}=4\cdot 3^6\cdot x^{20}\cdot y^{16}$

$\left(\dfrac{\color{red}{2a^4}}{\color{blue}{b^3}}\right)^{2}=$$\,\dfrac{\left(2a^4\right)^2}{\left(b^3\right)^2}=$$\,\dfrac{4a^8}{b^6}$

$\dfrac{a^{n}}{a^{m}} = a^{n-m}$

Exponential identities

$\dfrac{a^{\color{red}{5}}}{a^{\color{blue}{3}}}=\dfrac{{\color{red}{a}}\cdot{\color{red}{a}}\cdot{\color{red}{a}}\cdot{\color{red}{a}}\cdot{\color{red}{a}}}{{\color{blue}{a}}\cdot{\color{blue}{a}}\cdot{\color{blue}{a}}}=a^{{\color{red}{5}}-{\color{blue}{3}}}=a^{2}$

$\dfrac{5^3}{5^2} = \dfrac{5\cdot 5\cdot 5}{5\cdot 5} = 5^{3-2}=5$

$\dfrac{a^{n}}{a^{m}} = a^{n-m}$

$\dfrac{6{\color{red}{x}}^3{\color{blue}{y}}^5}{3{\color{red}{x}}^2{\color{blue}{y}}}=$ $\,2{\color{red}{x}}^{3-2}{\color{blue}{y}}^{5-1}=2xy^4$

$\left(\dfrac{a^4b^5}{a^2b}\right)^2=$$\,\left(a^2b^4\right)^2=a^4b^8$

$\left(\dfrac{x^5y^3z}{x^3y^2}\right)^2=$$\,\dfrac{x^{10}y^6z^2}{x^6y^4}=x^4y^2z^2$

Negative exponents

$$\bbox[8pt]{\Large{a^{\color{red}{-1}} = \dfrac{1}{a}}}$$

$\dfrac{5^2}{5^3}=\dfrac{5\cdot 5}{5\cdot 5\cdot 5}={\color{blue}{\dfrac{1}{5}}}=5^{2-3}={\color{blue}{5^{-1}}}$

$a^{-1}=\dfrac{1}{a}$

$\dfrac{{\color{red}{x}}^2{\color{blue}{y}}^3}{{\color{red}{x}}^3{\color{blue}{y}}^4}=$ $\,\dfrac{1}{{\color{red}{x}}{\color{blue}{y}}}={\color{red}{x}}^{2-3}{\color{blue}{y}}^{3-4}=({\color{red}{x}}{\color{blue}{y}})^{-1}$

$2xy^{-1}=$$\,2x\cdot\dfrac{1}{y}=\dfrac{2x}{y}$

$3(ab)^{-1}=$$\,3\cdot\dfrac{1}{ab}=\dfrac{3}{ab}$

$\left(\dfrac{a}{b}\right)^{-1}=\dfrac{b}{a}$

$\left(\dfrac{\color{red}{a}}{\color{blue}{b}}\right)^{-1}=\dfrac{1}{\frac{\color{red}{a}}{\color{blue}{b}}}=1:\dfrac{\color{red}{a}}{\color{blue}{b}}=1\cdot\dfrac{\color{blue}{b}}{\color{red}{a}}=\dfrac{\color{blue}{b}}{\color{red}{a}}$

$\left(\dfrac{\color{red}{2}}{\color{blue}{3}}\right)^{-1}=\dfrac{\color{blue}{3}}{\color{red}{2}}$

$\left(\dfrac{a}{b}\right)^{-1}=\dfrac{b}{a}$

$\left(\dfrac{2{\color{red}{x}}^2{\color{blue}{y}}^3}{{3\color{red}{x}}^5{\color{blue}{y}}^4}\right)^{-1}=$ $\,\dfrac{{3\color{red}{x}}^5{\color{blue}{y}}^4}{2{\color{red}{x}}^2{\color{blue}{y}}^3}=\dfrac{3}{2}x^3y$

$\left(\dfrac{5{\color{red}{a}}^7{\color{blue}{b}}^5}{{6\color{red}{a}}^5{\color{blue}{b}}^4}\right)^{-1}=$ $\,\left(\dfrac{{5\color{red}{a}}^2{\color{blue}{b}}}{6}\right)^{-1}=\dfrac{6}{5a^2b}$

$a^{-n}=\dfrac{1}{a^{n}}$

Exponential identities

$a^{-n}=\left(a^{\color{red}{-1}}\right)^n=\left(\dfrac{1}{a}\right)^n=\dfrac{1}{a^n}$

$(3x)^{-2}=\left(\dfrac{1}{3x}\right)^2=\dfrac{1^2}{(3x)^2}=\dfrac{1}{9x^2}$

$a^{-n}=\dfrac{1}{a^{n}}$

$\left(\dfrac{6{\color{red}{x}}^3{\color{red}{y}}^2}{2{\color{blue}{x}}^6{\color{blue}{y}}^3}\right)^{-2}=$ $\,\left(\dfrac{2{\color{blue}{x}}^6{\color{blue}{y}}^3}{6{\color{red}{x}}^3{\color{red}{y}}^2}\right)^{2}=\left(3x^3y\right)^2$

$\left(\dfrac{1}{2a^2b^3}\right)^{-3}=$$\,\left(\dfrac{2a^2b^3}{1}\right)^{3}=8a^3b^9$

$\left(5x^2y^{-3}\right)^{-2}=$$\,\left(\dfrac{5x^2}{y^3}\right)^{-2}=\dfrac{y^6}{25x^4}$

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