Indices
Indices, (also known as exponents or powers), are a way of representing how many times a number or variable (the base) is multiplied by itself. For example, in \(2^3\), 2 is the base, and 3 is the index (or exponent), meaning \(2 \times 2 \times 2 = 8\).
\(\text{So, } a \times a \times a \times a = a^4\)
Formula for Indices
Here are some key formulas for working with indices:
1. Multiplication Rule: \( a^m \times a^n = a^{m+n} \)
2. Division Rule: \( \frac{a^m}{a^n} = a^{m-n} \) (where \( a \neq 0 \))
3. Power of a Power Rule: \( (a^m)^n = a^{m \times n} \)
4. Power of a Product Rule: \( (ab)^n = a^n \times b^n \)
5. Zero Exponent Rule: \( a^0 = 1 \) (where \( a \neq 0 \))
6. Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)
7. \(n^th\) Root Rule: \( \sqrt[n]{a} = a^{\frac{1}{n}} \)
Example Questions with solutions
\(
\begin{aligned}
& \text{Q1. Find the value of: } 2^3 \times 2^5 \\
& \text{Solution:} \\
& = 2^3 \times 2^5 \\
& = 2^{(3+5)} \\
& \therefore 2^8 \text{ (Answer)}
\end{aligned}
\)
\(
\begin{aligned}
& \text{Q2. Find the value of: } (\frac{x^m}{x^{-n}})^{m^2-mn+n^2} \times (\frac{x^n}{x^{-l}})^{n^2-nl+l^2} \times (\frac{x^l}{x^{-m}})^{l^2-lm+l^2} \\
& \text{Solution:} \\
& = (\frac{x^m}{x^{-n}})^{m^2-mn+n^2} \times (\frac{x^n}{x^{-l}})^{n^2-nl+l^2} \times (\frac{x^l}{x^{-m}})^{l^2-lm+m^2} \\
& = x^{(m+n)(m^2-mn-n^2)} \times x^{(n+1)(n^2-nl+l^2)} \times x^{(l+m)(l^2-lm+m^2)} \\
& = x^{(m^3 + n^3)} \times x^{(n^3 + l^3)} \times x^{(l^3 + m^3)} \\
& = x^{(m^3 + n^3 + n^3 + l^3 +l^3 + m^3)} \\
& \therefore x^{2(l^3 + m^3 + n^3)} \text{ (Answer)}
\end{aligned}
\)
\(
\begin{aligned}
& \text{Q3. Prove: } \frac{1}{1 + x^{a-b} + x^{c-b}} + \frac{1}{1 + x^{b-c} + x^{a-c}} + \frac{1}{1 + x^{c-a} + x^{b-a}} = 1 \\
& \text{Solution:} \\
& \text{LHS:} \\
& = \frac{1}{1 + x^{a-b} + x^{c-b}} + \frac{1}{1 + x^{b-c} + x^{a-c}} + \frac{1}{1 + x^{c-a} + x^{b-a}} \\
& = \frac{1}{1 + \frac{x^a}{x^b} + \frac{x^c} {x^b}} + \frac{1}{1 + \frac{x^b}{x^c} + \frac{x^a}{x^c}} + \frac{1}{1 + \frac{x^c}{x^a} + \frac{x^b}{x^a}} \\
& = \frac{1}{\frac{x^b + x^a + x^c}{x^b}} + \frac{1}{\frac{x^c + x^b + x^a}{x^c}} + \frac{1}{\frac{x^a + x^c + x^b}{x^a}} \\
& = \frac{x^b}{x^a + x^b +x^c} + \frac{x^c}{x^a + x^b +x^c} + \frac{x^a}{x^a + x^b +x^c} \\
& = \frac{x^a + x^b +x^c}{x^a + x^b +x^c} \\
& = 1 =\text{RHS} \\
& \therefore \text{Hence Proved}
\end{aligned}
\)
Class 9: Mathematics
