INDICES

Definition of Indices

Indices or Index(singular) deals with numbers raised to some powers, example: X² which we call X raised to the power of 2.
In the above example, X is the Base while "²" is the power or index or quotient.
Indices can also be called Exponents.

Laws of Indices

1. Multiplication Law of Indices Says: X^a x X^b = X^a+b
2. Division Law of Indices Says: X^a÷ X^b or X^a X^b = X^a-b
3. Power of Zero Law of Indices Says: X⁰= 1
4. Negative Power Law of Indices Says: X^-a = 1 X^a
5. Bracket Law of Indices Says: (X^a)^b = X^{a x b}
6. Fractional Power Law of Indices Says: X ^{a b} = (^b√X)^a

Question 1

If 27^x x 3^1-x 9^2x = 1 , Find x

WAEC 2015

SOLUTION

Lets rewrite the question

27^x x 3^1-x 9^2x

We have to first take the LHS to the same base

27 = 3 x 3 x3 = 3³

9 = 3 x 3 = 3²

3 = 3 = 3

So,

3^3(x) x 3^(1-x) 3^2(2x) = 1

3^3x x 3^1-x 3^4x = 1

From the multiplication Law of Indices, we will add the powers,

3^{3x + (1-x)} 3^4x = 1

3^{3x + 1-x} 3^4x = 1

Collecting Live terms of the powers.

3^{2x + 1} 3^4x = 1

We then have to apply the division Law of Indices, By this we mean to subtract the powers of the denominator since its same base (3).

3^{2x + 1 - (4x)} = 1

Then we expand the bracket, we have

3^{2x + 1 -4x} = 1

Collect the like terms, we have

3^{-2x + 1} = 1

Then we apply the zero law to the RHS
Recall that anything raise to power of zero is 1.

3^{-2x + 1} = 3⁰

Equating the powers

-2𝑥+1 = 0

Collecting like terms
-2𝑥 = 0 - 1
-2𝑥 = - 1
Hence we have - on both side, we take care of it.
Making the equation 2𝑥 = 1
Then making 𝑥 the subject of the formular,
We have
𝑥 = 1 2 Answer