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 Simultaneous Linear Equations « PreviousNext » Logarithms 
Author: S.LAL
Created: 22 Aug, 2010; Last Modified: 29 Apr, 2018

Algebra II - 02

Indices (Exponents)

If n is a positive integer, then by definition, an stands for a being multiplied with itself n times, i.e. a × a × a × a × ... upto n terms. Here, a is the base and n is called the power or exponent or index (plural indices).

Case: zero exponent

When the value of the power is zero (n = 0), we have the identity:

a 0 = 1
.......(1)

This implies that any quantity with a zero exponent has a value of 0.

Thus, 5 0 = 34 0 = b 0 = 1 .

Case: negative exponent

When the value of the exponent is negative (-n, where n is a positive integer), we have:

a − n ≡ 1 a n
.......(2)

a − n is known as the reciprocal of a n . (The symbol ≡ means "identically equal to").

Thus, if n is a positive integer, a − n stands for the reciprocal of a being multiplied with itself n times, i.e. 1 a × 1 a × 1 a × 1 a × ... upto n terms.

In other words, a number raised to a negative exponent is the reciprocal of that number raised to a positive exponent.

The corollary of Eq. is:

a n ≡ 1 a − n
.......(3)

This further implies that any factor with an exponent can be relocated from the numerator to the denominator, or vice-versa, by changing the sign of the exponent.

Thus, 3 x 2 y 5 ⇔ 3 x 2 y − 5 ⇔ 3 y 5 x − 2 ⇔ 3 x − 2 y − 5 ⇔ 1 3 − 1 x − 2 y − 5

Case: fractional exponent

When the value of the exponent is fractional (of form p q , where p and q are positive integers), we have:

a p / q ≡ a p q
.......(4)

The above means that a raised to the power of p q is identically equal to the qth root of a raised to the power p.

In other words, when a number is raised to a fractional power, the exponent of the number is the numerator of the fraction and the root of the number is the denominator of the fraction.

As a corollary of Eq.

a 1 / q ≡ a q
.......(5)

Thus,

4 1 / 2 = 4 1 2 = 4 2 = ± 2

(Note: the symbol 2 for square root is also written as just )

8 3 / 2 = 8 2 3 = 8 × 8 3 = 8 3 × 8 3 = 2 × 2 = 4

Laws of Indices

The following laws hold for all integral (positive and negative) and fractional values of m and n, as long there is no scope for division by 0 anywhere.

Product Law

When two numbers having the same base are multiplied, the indices are added.

a m × a n = a m + n
.......(6)

Evaluate (i) 2 3 × 2 2 (ii) 2 3 / 2 × 2 1 / 2 .

(i) 2 3 × 2 2 = 2 3 + 2 = 2 5 = 32

(ii) 2 3 / 2 × 2 1 / 2 = 2 3 2 + 1 2 = 2 4 2 = 2 2 = 4

Quotient Law

When a number is divided by another having the same base, the indices are subtracted.

a m a n = a m − n ( a ≠ 0 )
.......(7)

Evaluate (i) 3 4 3 2 (ii) 3 3 / 2 3 1 / 2 .

(i) 3 4 3 2 = 3 4 − 2 = 3 2 = 9

(ii) 3 3 / 2 3 1 / 2 = 3 3 2 − 1 2 = 3 2 2 = 3 1 = 3

Power Law

When a number raised to a power is further raised to a power, the indices are multiplied.

( a m ) n = a m × n
.......(8)

Evaluate (i) 4 3 × 8 2 (ii) 8 5 / 4 4 3 / 2 .

(i) 4 3 × 8 2 = ( 2 2 ) 3 × ( 2 3 ) 2 = 2 2 × 3 × 2 3 × 2 = 2 6 × 2 6 = 2 6 + 6 = 2 12 = 4096

(ii) 8 5 / 6 4 3 / 2 = ( 2 3 ) 5 / 6 ( 2 2 ) 3 / 2 = 2 ( 3 × 5 6 ) 2 ( 2 × 3 2 ) = 2 5 2 2 3 = 2 5 2 − 3 = 2 5 − 6 2 = 2 − 1 2 = 1 2 1 / 2 = 1 2

Given 1176 = 2 p ⋅ 3 q ⋅ 7 r , find (i) the numerical values of p, q and r (ii) the value of 2 p ⋅ 3 q ⋅ 7 − r as a fraction. (Bansal, Mathematics Class IX, p. 109 Ex. 3)

Expressing 1176 as factors (found by successive division):

1176 ≡ 2 × 2 × 2 × 3 × 7 × 7 ≡ 2 3 ⋅ 3 1 ⋅ 7 2

This implies that

2 3 ⋅ 3 1 ⋅ 7 2 = 2 p ⋅ 3 q ⋅ 7   r

(i) Equating the exponents of the same bases, we find that p = 3; q = 1; r = 2

(ii) 2 p ⋅ 3 q ⋅ 7   − r = 2 3 ⋅ 3 1 ⋅ 7   − 2 = 2 3 ⋅ 3 1 7 2 = 8 ⋅ 3 49 = 24 49

Find the value of x, if (   3 5   ) 1 − 2 x = 4 17 27 . ICSE 1991

( 3 5 ) 1 − 2 x = 4 17 27 ⇒ ( 3 5 ) 1 − 2 x 2 = ( 4 ⋅ 27 ) + 17 27 = 125 27 ⇒ ( 3 5 ) 1 − 2 x 2 = 5 3 3 3 = 3 − 3 5 − 3 = ( 3 5 ) − 3

From the above, once the base (i.e. 3 5 ) are the same on both sides of the equation, we can equate the exponents, so that,

1 − 2 x 2 = − 3 ⇒ 1 − 2 x = − 6 ⇒ − 2 x = − 7 ⇒ x = 7 2

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List of References

Bansal, RK, Concise Mathematics I.C.S.E., Part I – Class IX, New Delhi: Selina Publishers, 2005.

Bibliography

Bird, J, Basic Engineering Mathematics, 4th edn, Oxford, UK: Elsevier, 2005.
Gupta, SD & Banerjee, A, ICSE Mathematics for Class 9, Patna, India: Bharati Bhawan, 2003.
Hall, HS & Knight, SR, Elementary Algebra for Schools, metric edn, Agra, India: AK Publications, 1966.

 Simultaneous Linear Equations « PreviousNext » Logarithms 
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