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Laws and Properties of Arithmetics : Exponents

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Overview

Exponents :

»  Exponent is a form of multiplication
→  ${a}^{m}=a×a×a...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$${a}^{m} = a \times a \times a \ldots \left(m \textrm{\times}\right)$

»  Procedural Simplifications Derived from Properties of Multiplication
→  ${a}^{m}+{a}^{m}+...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)=n{a}^{m}$${a}^{m} + {a}^{m} + \ldots \left(n \textrm{\times}\right) = n {a}^{m}$
→  $p{a}^{m}+q{a}^{m}=\left(p+q\right){a}^{m}$$p {a}^{m} + q {a}^{m} = \left(p + q\right) {a}^{m}$
→  ${a}^{m}×{a}^{n}={a}^{m+n}$${a}^{m} \times {a}^{n} = {a}^{m + n}$
→  ${a}^{m}×{b}^{m}={\left(a×b\right)}^{m}$${a}^{m} \times {b}^{m} = {\left(a \times b\right)}^{m}$
→  ${a}^{m}÷{a}^{n}={a}^{m-n}$${a}^{m} \div {a}^{n} = {a}^{m - n}$
→  ${a}^{m}÷{b}^{m}={\left(a÷b\right)}^{m}$${a}^{m} \div {b}^{m} = {\left(a \div b\right)}^{m}$
→  ${\left({a}^{m}\right)}^{n}={a}^{mn}$${\left({a}^{m}\right)}^{n} = {a}^{m n}$
→  ${a}^{\frac{1}{m}}=\sqrt[m]{a}$${a}^{\frac{1}{m}} = \sqrt[m]{a}$
→  ${a}^{-m}=\frac{1}{{a}^{m}}$${a}^{- m} = \frac{1}{{a}^{m}}$

»  Some forms cannot be simplified any further
the expressions can be evaluated to equivalent numerical values, but not simplified retaining the power of $a$$a$
→  $p{a}^{m}+q{a}^{n}+r$$p {a}^{m} + q {a}^{n} + r$ cannot be simplified for $m\ne n$$m \ne n$
→  ${\left(a+b\right)}^{m}$${\left(a + b\right)}^{m}$ cannot be simplified, but expanded
The first is the basis of algebraic expressions and polynomials, and the second is the basis of algebraic identities.

results from exponents

Exponent is defined as repeated multiplication.
$p×p×p...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)={p}^{m}$$p \times p \times p \ldots \left(m \textrm{\times}\right) = {p}^{m}$

${p}^{m}×{p}^{n}={\left(p\right)}^{m+n}$${p}^{m} \times {p}^{n} = {\left(p\right)}^{m + n}$.

${p}^{m}×{p}^{n}$${p}^{m} \times {p}^{n}$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$×p×p×...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)$$\times p \times p \times \ldots \left(n \textrm{\times}\right)$

$=p×p×...\left(m+n\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m + n \textrm{\times}\right)$

$={\left(p\right)}^{m+n}$$= {\left(p\right)}^{m + n}$

${p}^{m}×{q}^{m}={\left(p×q\right)}^{m}$${p}^{m} \times {q}^{m} = {\left(p \times q\right)}^{m}$.

${p}^{m}×{q}^{m}$${p}^{m} \times {q}^{m}$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$×q×q×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$\times q \times q \times \ldots \left(m \textrm{\times}\right)$

$=pq×pq×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p q \times p q \times \ldots \left(m \textrm{\times}\right)$

$={\left(pq\right)}^{m}$$= {\left(p q\right)}^{m}$

${p}^{m}÷{p}^{n}={p}^{m-n}$${p}^{m} \div {p}^{n} = {p}^{m - n}$

${p}^{m}÷{p}^{n}$${p}^{m} \div {p}^{n}$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$÷\left(p×p×p...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$$\div \left(p \times p \times p \ldots \left(n \textrm{\times}\right)\right)$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$÷p÷p÷p...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$

$=p×p×...\left(m-n\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m - n \textrm{\times}\right)$

$={\left(p\right)}^{m-n}$$= {\left(p\right)}^{m - n}$

${p}^{m}÷{q}^{m}={\left(p÷q\right)}^{m}$${p}^{m} \div {q}^{m} = {\left(p \div q\right)}^{m}$

${p}^{m}÷{q}^{m}$${p}^{m} \div {q}^{m}$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$÷\left(q×q×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$$\div \left(q \times q \times \ldots \left(m \textrm{\times}\right)\right)$

$=p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \textrm{\times}\right)$$÷q÷q÷...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$

$=\left(p÷q\right)×\left(p÷q\right)×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= \left(p \div q\right) \times \left(p \div q\right) \times \ldots \left(m \textrm{\times}\right)$

$={\left(p÷q\right)}^{m}$$= {\left(p \div q\right)}^{m}$

${p}^{0}=1$${p}^{0} = 1$.

${p}^{0}={p}^{1-1}=\frac{p}{p}=1$${p}^{0} = {p}^{1 - 1} = \frac{p}{p} = 1$

${\left({p}^{m}\right)}^{n}={p}^{mn}$${\left({p}^{m}\right)}^{n} = {p}^{m n}$

${\left({p}^{m}\right)}^{n}$${\left({p}^{m}\right)}^{n}$

$={p}^{m}×{p}^{m}×...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$

$=\left(p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$$= \left(p \times p \times \ldots \left(m \textrm{\times}\right)\right)$$×...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)\right)$

$=p×p×...\left(m×n\phantom{\rule{1ex}{0ex}}\text{times}\right)$$= p \times p \times \ldots \left(m \times n \textrm{\times}\right)$

$={\left(p\right)}^{mn}$$= {\left(p\right)}^{m n}$

$\sqrt[m]{p}={p}^{\frac{1}{m}}$$\sqrt[m]{p} = {p}^{\frac{1}{m}}$

Root is one of the inverses of exponent: For the exponent $p={q}^{m}$$p = {q}^{m}$, one inverse is $q=\sqrt[m]{p}={p}^{\frac{1}{m}}$$q = \sqrt[m]{p} = {p}^{\frac{1}{m}}$

${p}^{-m}=\frac{1}{{p}^{m}}$${p}^{- m} = \frac{1}{{p}^{m}}$

${p}^{-m}$${p}^{- m}$

$=\frac{1}{p×p×...\left(m\phantom{\rule{1ex}{0ex}}\text{times}\right)}$$= \frac{1}{p \times p \times \ldots \left(m \textrm{\times}\right)}$

$=\frac{1}{{p}^{m}}$$= \frac{1}{{p}^{m}}$

${p}^{m}+{p}^{m}+\dots \left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)=n{p}^{m}$

$k×{p}^{m}+l×{p}^{m}=\left(k+l\right)×{p}^{m}$$k \times {p}^{m} + l \times {p}^{m} = \left(k + l\right) \times {p}^{m}$

$k×{p}^{m}+l×{p}^{m}$$k \times {p}^{m} + l \times {p}^{m}$

$={p}^{m}+{p}^{m}+\dots \left(k\phantom{\rule{1ex}{0ex}}\text{times}\right)$$+{p}^{m}+{p}^{m}+\dots \left(l\phantom{\rule{1ex}{0ex}}\text{times}\right)$

$={p}^{m}+{p}^{m}+{p}^{m}+{p}^{m}+\dots \left(\left(k+l\right)\phantom{\rule{1ex}{0ex}}\text{times}\right)$

$=\left(k+l\right)×{p}^{m}$$= \left(k + l\right) \times {p}^{m}$

It is noted that $k×{p}^{m}+l×{p}^{n}$$k \times {p}^{m} + l \times {p}^{n}$ cannot be simplified. Expressions of this type lead to the definition of algebraic expressions and polynomials in the form $a{x}^{m}+b{x}^{n}+c$$a {x}^{m} + b {x}^{n} + c$.

It is noted that ${\left(p+q\right)}^{m}$${\left(p + q\right)}^{m}$ cannot be simplified. Expressions of this type lead to the definition of algebraic identities.

summary

Properties of Exponents
${a}^{m}+{a}^{m}+...\left(n\phantom{\rule{1ex}{0ex}}\text{times}\right)=n{a}^{m}$${a}^{m} + {a}^{m} + \ldots \left(n \textrm{\times}\right) = n {a}^{m}$
$p{a}^{m}+q{a}^{m}=\left(p+q\right){a}^{m}$$p {a}^{m} + q {a}^{m} = \left(p + q\right) {a}^{m}$
${a}^{m}×{a}^{n}={a}^{m+n}$${a}^{m} \times {a}^{n} = {a}^{m + n}$
${a}^{m}×{b}^{m}={\left(a×b\right)}^{m}$${a}^{m} \times {b}^{m} = {\left(a \times b\right)}^{m}$
${a}^{m}÷{a}^{n}={a}^{m-n}$${a}^{m} \div {a}^{n} = {a}^{m - n}$
${a}^{m}÷{b}^{m}={\left(a÷b\right)}^{m}$${a}^{m} \div {b}^{m} = {\left(a \div b\right)}^{m}$
${\left({a}^{m}\right)}^{n}={a}^{mn}$${\left({a}^{m}\right)}^{n} = {a}^{m n}$
${a}^{\frac{1}{m}}=\sqrt[m]{a}$${a}^{\frac{1}{m}} = \sqrt[m]{a}$
${a}^{-m}=\frac{1}{{a}^{m}}$${a}^{- m} = \frac{1}{{a}^{m}}$
${\mathrm{log}}_{a}{a}^{m}=m$${\log}_{a} {a}^{m} = m$

•  Some forms cannot be simplified any further
That is, the expressions can be evaluated to equivalent numerical values, but not simplified retaining the power of $a$$a$

→  $p{a}^{m}+q{a}^{n}+r$$p {a}^{m} + q {a}^{n} + r$ cannot be simplified for $m\ne n$$m \ne n$
This is the basis for algebraic expressions and polynomials

→  ${\left(a+b\right)}^{m}$${\left(a + b\right)}^{m}$ cannot be simplified, but an equivalent expression can be defined in the general form.
This is the basis for algebraic identities.

Outline

The outline of material to learn "Algebra Foundation" is as follows.

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations