Indices

Which is the greatest number in the following?

(3)⁴, (9)³, (81)²

Here, the bases of all three terms and different. So it is difficult to compare. Therefore, converting all bases into the same base;

9 = 3²

\(\therefore\) 9³ = (3²)³ = 3^2×3 = 3⁶

Similarly, 81 = 3⁴

\(\therefore\) (81)² = (3⁴)² = 3^4×2 = 3⁸

Now, 3 is the base in 3⁴, 3⁶, 3⁸. Therefore, the number having the greatest exponent with the same base is the greatest number.

∴ 3⁸ or (81)² is the greatest number.

Let us recall the laws of indices which we have studied in the previous classes.

1. Law of zero index: x⁰ = 1 

2. Product law of indices: x^m \(\times\) xⁿ = x ^m+n , (power are added in multiplication of same bases)

3. Power law of indices: (x^m)ⁿ = x m\(\times\)h

4. Law of negative index: x^-m = \(\frac{1}{x^m}\)

5. Root law of indices: x\(\frac pq\) = \(\sqrt[q]{x^p}\) = (\(\sqrt[q]{x}\))^p

6. Quotient law of indices: a^m ÷ aⁿ = \(\frac{a^m}{a^n}\) = \(\frac{1}{a^{(n-m)}}\)

(\(\frac{x}{y}\))ⁿ= \(\frac{x^n}{y^n}\)

7. (xy) ^m = x^m y^m

8. \(\sqrt[n]{x}\) = x^{\(\frac 1n\)} 

These rules are known as laws of indices.

Example:

Find the value of: \(\sqrt[3]{8^2}\)

\(\sqrt[3]{8^2}\)

= 8^{\(\frac{2}{3}\)}

= 2^{3×\(\frac{2}{3}\)}

= 2²

= 4_Ans

Exponential equation:

Exponential equation is an algebraic equation where unknown variables appears as an exponent of a base. We equate the power if the base on both sides of a equation is equal. We use the following rules while solving exponential equation:

1. If a^x = a^y then x = y

2. If a^x = 1, then a^x= a⁰

∴ x = 0

3. If a^x = k¹, then a = k^{\(\frac 1x\)}

4. If a^x = b^y, then a = b^{\(\frac yx\)}