\displaystyle \textbf{EXPONENTIAL FORM}

\displaystyle \text{Exponentiation is a mathematical operation in which a number is raised}
\displaystyle \text{to a power. An expression of the form }b^n\text{ is called an exponential}
\displaystyle \text{expression, where }b\text{ is the base and }n\text{ is the exponent or index.}

\displaystyle b^n

\displaystyle \text{Here, }b\text{ is the base and }n\text{ is the exponent.}

\displaystyle \text{When }n\text{ is a positive integer, }b^n\text{ means that }b\text{ is multiplied}
\displaystyle \text{by itself }n\text{ times.}

\displaystyle b^n=\underbrace{b\times b\times b\times\cdots\times b}_{n\text{ factors}}

\displaystyle \text{For example,}

\displaystyle 2^5=2\times2\times2\times2\times2=32.

\displaystyle \text{The expression }b^n\text{ is read as ``}b\text{ raised to the power }n\text{''}
\displaystyle \text{or ``the }n^{\text{th}}\text{ power of }b\text{''.}
\displaystyle \\

\displaystyle \textbf{ZERO AND NEGATIVE EXPONENTS}

\displaystyle \text{For any non-zero number }x,

\displaystyle x^0=1.

\displaystyle \text{For any positive integer }m,

\displaystyle x^{-m}=\frac{1}{x^m},\qquad x\neq0.

\displaystyle \text{Thus, a negative exponent indicates the reciprocal of the corresponding}
\displaystyle \text{positive power.}

\displaystyle \text{For example,}\qquad 2^{-3}=\frac{1}{2^3}=\frac{1}{8}.

\displaystyle \text{Also,}\qquad \frac{1}{x^{-m}}=x^m,\qquad x\neq0.
\displaystyle \\

\displaystyle \textbf{POWERS OF RATIONAL NUMBERS}

\displaystyle \text{If }\frac{p}{q}\text{ is a rational number and }m\text{ is a positive integer,}
\displaystyle \text{then, for }q\neq0,

\displaystyle \left(\frac{p}{q}\right)^m=\frac{p^m}{q^m}.

\displaystyle \text{The reciprocal of }\frac{p}{q}\text{ is }\frac{q}{p}, \text{ provided }p\neq0\text{ and }q\neq0.

\displaystyle \text{For example,}\qquad \left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}.
\displaystyle \\

\displaystyle \textbf{LAWS OF EXPONENTS}

\displaystyle \text{Let }x\text{ and }y\text{ be non-zero real numbers and let }m\text{ and }n
\displaystyle \text{be integers for which the following expressions are defined.}

\displaystyle \text{1. Product of powers with the same base:}

\displaystyle x^m\times x^n=x^{m+n}.

\displaystyle \text{For example,}\qquad x^3\times x^4=x^{3+4}=x^7.

\displaystyle \text{2. Quotient of powers with the same base:}

\displaystyle \frac{x^m}{x^n}=x^{m-n},\qquad x\neq0.

\displaystyle \text{If }m<n,\text{ then}

\displaystyle \frac{x^m}{x^n}=x^{m-n} =x^{-(n-m)}=\frac{1}{x^{n-m}}.

\displaystyle \text{For example,}\qquad \frac{x^3}{x^5}=x^{-2}=\frac{1}{x^2}.

\displaystyle \text{3. Power of a power:}

\displaystyle (x^m)^n=x^{mn}.

\displaystyle \text{For example,}\qquad (x^3)^4=x^{12}.

\displaystyle \text{4. Power of a product:}

\displaystyle (xy)^m=x^my^m.

\displaystyle \text{For example,}\qquad (2xy)^3=8x^3y^3.

\displaystyle \text{5. Power of a quotient:}

\displaystyle \left(\frac{x}{y}\right)^m =\frac{x^m}{y^m},\qquad y\neq0.

\displaystyle \text{For example,}\qquad \left(\frac{2x}{3y}\right)^2=\frac{4x^2}{9y^2}.

\displaystyle \text{6. Zero exponent:}

\displaystyle x^0=1,\qquad x\neq0.

\displaystyle \text{7. Negative exponent:}

\displaystyle x^{-m}=\frac{1}{x^m},\qquad x\neq0.
\displaystyle \\

\displaystyle \textbf{EQUALITY OF POWERS WITH THE SAME BASE}

\displaystyle \text{If two powers have the same positive base other than }1\text{, then equal}
\displaystyle \text{powers imply equal exponents.}

\displaystyle x^m=x^n\quad\Rightarrow\quad m=n, \qquad x>0,\quad x\neq1.

\displaystyle \text{For example,}\qquad 3^{2x}=3^8\quad\Rightarrow\quad2x=8\quad\Rightarrow\quad x=4.
\displaystyle \\

\displaystyle \textbf{RATIONAL EXPONENTS}

\displaystyle \text{For }x>0,\text{ and positive integers }m\text{ and }n,

\displaystyle x^{\frac{m}{n}} =\left(x^m\right)^{\frac{1}{n}} =\left(x^{\frac{1}{n}}\right)^m =\sqrt[n]{x^m} =\left(\sqrt[n]{x}\right)^m.

\displaystyle \text{In particular,}

\displaystyle x^{\frac{1}{n}}=\sqrt[n]{x}.

\displaystyle \text{For example,}\qquad 8^{\frac{2}{3}} =\left(\sqrt[3]{8}\right)^2 =2^2=4.
\displaystyle \\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.