Concept of Indices with Solved Examples

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We know that 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2⁶We know that 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2⁶

Here 2 is called the base and 6 is called the power (or index or exponent).

We say that "64 is equal to base 2 raised to the power 6".

Similarly, if m is a positive integer and

then a a a …m times = a^m.

Laws of Indices

If m and n are positive integers, and then

(i) a^m aⁿ = a^{m + n} [Product Law]

(ii) [Quotient Law]

(iii) (a^m)ⁿ = a^mn [Power Law]

(iv) (ab)^m = a^m . b^m

(v)

(vi) a^o = 1

(vii)

Surd

An irrational root of a positive rational number is called a surd. Consider a number with base 'a' as a positive rational number with power of a fraction, say then

Since is an n^th root, it is called a surd of order n, if it is irrational.

e.g., (i) is a surd of order 3.

(ii) is a surd of order 2.

(iii) is NOT a surd because and 3 is NOT an irrational number.

Without using tables, simplify:

(i) 36^-1/2 (ii)

(i) 36^-1/2

= (6²)^-1/2 = 6^-1

(ii)

Evaluate:

Given expression

Simplify:

(i)

(ii) If 2^x+2 = 128, find the value of x.

(iii) Simplify:

(iv) Simplify:

(v) Show that (x^{p - q})^{p + q} (x^{q - r})^{q + r} (x^{r - s})^r+s = 1

(vi) 49 7^x = (343)^{2x - 5} find 'x'.

(i) Given expression:

(ii) Since 128 = 2 2 2 2 2 2 2

= 2⁷

We have 2^x+2 = 2⁷ [ bases are equal ]

x + 2 = 7 [ powers are equal ]

x = 5

(iii)

(iv)

= (x^3a y⁶)^1/4 (x^2/3 y^-1)^a

= x^3a/4 y^3/2 x^2a/3 y^-a [ (x^3a)^1/4 = x^3a/4 (power law)]

(v) LHS = x^{(p - q) (p + q)} x^{(q - r) (q + r)} x^{(r - s) (r +s)}

= x⁰ = 1 = R.H.S

(vi) 49 7^x = (343)^{2x - 5}

7² 7^x = (7³)^{2x - 5}

7^2+x = 7^{6x - 15}

As the bases are equal, the powers are also equal.

2 + x = 6x - 15

or 5x = 17