JEE Mathematics Surds and Indices Notes Set 02

Index Notation

Repeated multiplication is written with an index: \( a\times a\times\cdots\times a \) (n factors) \( =a^n \). Here a is the base and n is the index or exponent. For example \( aaaabb=a^4b^2 \), and \( (x+y)(x+y)(x+y)=(x+y)^3 \). Check an identity by substituting numbers: for x = 1, y = -2 both sides of \( (x+y)^3=x^3+3x^2y+3xy^2+y^3 \) equal -1.

Laws of Indices

These laws hold for all integer and fractional indices, positive or negative, as long as the base is positive (or the root is defined). A common mistake is to add indices in a power of a power: \( (2^3)^4=2^{12} \), not \( 2^7 \). The laws apply only to the same base; to combine different bases, first write them with a common base.

Worked Examples

1. \( 64^{2/3}\times64^{3/2}=64^{2/3+3/2}=64^{13/6}=(2^6)^{13/6}=2^{13}=8192 \).
2. Simplify \( \dfrac{[5^3(2^3)^2 6^2]^3}{[5^7 3^2 2^{12}]^2} \): the bracket on top is \( 5^3 2^8 3^2 \); cubing gives \( 5^9 2^{24} 3^6 \); the bottom is \( 5^{14} 3^4 2^{24} \). The result is \( \frac{3^2}{5^5}=\frac{9}{3125} \).
3. \( (ab^2c^3)(bc^2a^3)^2(ca^2b^2)^3=a^{1+6+6}b^{2+2+6}c^{3+4+3}=a^{13}b^{10}c^{10} \).
4. If x, y, z are positive integers with \( x=y^z,\ y=z^x,\ z=x^y \): substituting gives \( x=x^{xyz} \), so xyz = 1 and x = y = z = 1.

Solving Exponential Equations

Write both sides as powers of the same base and equate the indices. Example: \( 2^x=8^{y+1} \) and \( 9^y=3^{x-9} \) give \( x=3y+3 \) and \( 2y=x-9 \), so y = 6 and x = 21. To compare roots such as \( 2^{1/2},3^{1/3},6^{1/6} \), raise all to the L.C.M. of the denominators (6): the values 8, 9, 6 show that \( 3^{1/3} \) is largest.