Power and Index | Quantitative Techniques for CLAT PDF Download

In the expression pⁿ, p is called the base and n is called the power or index.

For example, in (-5)⁷, -5 is the base and 7 is the power.

In the following expressions the placeholders show example bases and powers:

In the expression above, the element shown in

is the base and 7 is the power.

=

Remarks

In exponential notation, the base may be any rational number and the power (index) may be any integer.

Note

If the power of a rational number is 1, its value equals the rational number itself.

For example, (-10)¹ = -10.

In general,

Example

Find the value of

Solution:

Laws of Exponents (Indices)

The following laws are valid for a non-zero rational base p and integers (positive, zero, or negative) m, n, r, s where indicated.

• Product law: p^m × pⁿ = p^m+n.
• The product law extends: p^m × pⁿ × p^r × p^s = p^m+n+r+s.
• Power of a power: (p^m)ⁿ = p^mn.
• Quotient law: p^m ÷ pⁿ = p^m-n for m > n.
• For m < n, p^m ÷ pⁿ=
  
  .
• Zero power: For p ≠ 0, p⁰ = 1.
• Reciprocal (power -1): For p ≠ 0, p^-1 = 1/p. Thus (p)^-1=
  
  .
• Negative integer power: p^-m=
  
  .
• Product with same exponent: p^m × q^m = (p × q)^m.
• Other related identities are shown below:
  

Worked examples showing application of laws of exponents

Example 1

Simplify (a)

Solution:

= 2^-2 × 3^-4+2 = 2^-2 × 3^-2

Example 2

Find m if

Solution:

(a)

LHS =

RHS =

Equating LHS and RHS

Because the bases are the same, the powers must be equal.

So -2m + 1 = -27

or -2m = -27 - 1

= -28

or m = 14

(b)

LHS =

RHS = 2^m

So 2^m = 2⁵

or m = 5

Example

Solve for x

a) 3^x = 81

b) (7^2x)^-2 = (2401)^-1

Solution

a)

RHS = 81 = 3⁴

So 3^x = 3⁴

Therefore x = 4

b)

RHS = (2401)^-1 = (7⁴)^-1 = 7^-4

LHS = (7^2x)^-2 = 7^-4x

Equating LHS and RHS

7^-4x = 7^-4

Since bases are equal, -4x = -4

Or x = 1

Understanding differences between similar notations

The following comparisons clarify different exponent notations and their meanings.

= p^m × p^m × p^m × p^m ... (n times)

= p^{m + m + m + ...} (n times)

= p^m·n

Whereas

=

Let us illustrate the difference with a concrete example.

Find the difference between

(2²)³ = 2² × 2² × 2² = 2⁶

= 2^2×2×2 = 2⁸

This example shows the distinction: applying an exponent to a product of repeated groups (as in repeated multiplication of whole groups) is different from raising an already exponentiated quantity to another power. One case gives power equal to product of exponents, the other depends on how the expression is grouped. Pay attention to parentheses and order of operations when working with powers.

Summary

• In pⁿ, p is the base and n is the power (index).
• Key exponent laws: product law, quotient law, power of a power, zero power, negative powers and reciprocal rules are used to simplify expressions and solve equations involving exponents.
• Always check bases and use equality of powers only when bases are equal and non-zero.
• Use parentheses carefully; the position of parentheses changes the meaning and result of exponentiation.