We looked at exponents in Class 8 as well. Before starting this topic, related to Class 9, it is a good idea to quickly revisit what we learned in the previous class.

Class 8: Exponents – Lecture Notes

Class 8: Exponents – Exercise 18

Indices : For any real number \displaystyle a and positive integer \displaystyle n, we define
\displaystyle a\times a\times a\times\cdots\text{ (to }n\text{ factors)}=a^n
Here, \displaystyle a is called the base and \displaystyle n is called the index or exponent.

**Some Laws of Indices**

\displaystyle \text{(i) }a^0=1

\displaystyle \text{(ii) }a^{-n}=\frac{1}{a^n}

\displaystyle \text{(iii) }a^m\times a^n=a^{m+n}

\displaystyle \text{(iv) }a^m\div a^n=a^{m-n}

\displaystyle \text{(v) }\left(a^m\right)^n=a^{mn}

\displaystyle \text{(vi) }a^{-m}=\frac{1}{a^m}

\displaystyle \text{(vii) }(ab)^m=a^m\cdot b^m

\displaystyle \text{(viii) }\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}

\displaystyle \text{(ix) }\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n

\displaystyle \text{(x) If }a^m=a^n,\text{ then }m=n,\text{ where }a>0\text{ and }a\ne1.

\displaystyle \text{(xi) If }p^m\times q^n\times r^o=p^aq^br^c,\text{ then }m=a,\ n=b\text{ and }o=c,\text{ where }p,\ q\text{ and }r\text{ are different primes.}

\displaystyle \text{(xii) If }a^m=b^m,\text{ then }a=b,\text{ where }a\text{ and }b\text{ are positive.}

\displaystyle \text{(xiii) }\sqrt{a}=a^{\frac12},\qquad \sqrt[3]{a}=a^{\frac13},\qquad \sqrt[4]{a}=a^{\frac14}

**Some formulae which are used**

\displaystyle \text{(i) }(a+b)(a-b)=a^2-b^2

\displaystyle \text{(ii) }(a-b)(a^2+ab+b^2)=a^3-b^3

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

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