List of Formulas on Indices

List of Formulas on Indices

In mathematics, indices (also called exponents or powers) play a crucial role in simplifying large numbers, solving algebraic equations, and working with exponential functions. The laws of indices form the foundation of algebra, higher mathematics, and competitive exams. In this post, we will cover all important formulas on indices, along with rules, properties, and examples.

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What are Indices?

Indices represent repeated multiplication of a number by itself. For example, \(4^3 = 4 \times 4 \times 4 = 64\). Here, 4 is the base, and 3 is the index or exponent. Indices help us express very large or very small numbers in a compact form.

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1. Basic Laws of Indices

The following are the fundamental laws of indices:

• Product Law: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Law: \(\frac{a^m}{a^n} = a^{m-n}, \; a \neq 0\)
• Power Law: \((a^m)^n = a^{mn}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)
• Power of a Quotient: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \; b \neq 0\)

Example: \(2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128\)

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2. Zero Index Law

• \(a^0 = 1, \; a \neq 0\)

Example: \(5^0 = 1\)

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3. Negative Index Law

• \(a^{-n} = \frac{1}{a^n}, \; a \neq 0\)

Example: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)

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4. Fractional Indices

• \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
• \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m\)

Example: \(16^{\frac{1}{2}} = \sqrt{16} = 4\)

Example: \(27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9\)

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5. Important Properties of Indices

• \(a^1 = a\)
• \((a^m)^0 = 1, \; a \neq 0\)
• If \(a^x = a^y\) (where \(a > 0, a \neq 1\)), then \(x = y\)
• If \(a^m = b^m, \; (m \neq 0)\), then \(a = b\)
• If \(x^m = x^n\), then \(m = n\) (provided \(x > 0\) and \(x \neq 1\))

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6. Alternative Forms of Indices Laws

• \(\frac{a^m}{a^n} = a^{m-n} = \frac{1}{a^{n-m}}\)
• \(a^{\frac{-m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}\)
• \((a^{-1})^n = \frac{1}{a^n}\)

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7. Examples Using Laws of Indices

Example 1: Simplify \(3^4 \cdot 3^2\)
\(= 3^{4+2} = 3^6 = 729\)

Example 2: Simplify \(\frac{7^5}{7^2}\)
\(= 7^{5-2} = 7^3 = 343\)

Example 3: Simplify \((2^3)^2\)
\(= 2^{3 \cdot 2} = 2^6 = 64\)

Example 4: Simplify \(25^{-\frac{1}{2}}\)
\(= \frac{1}{25^{\frac{1}{2}}} = \frac{1}{\sqrt{25}} = \frac{1}{5}\)

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8. Applications of Indices

Indices are widely used in:

• Simplifying large algebraic expressions
• Scientific notation (expressing very large or very small numbers)
• Exponential growth and decay problems
• Compound interest calculations
• Physics (laws involving powers and exponents)

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Frequently Asked Questions (FAQs) on Indices

Q1. What are the 5 laws of indices?

The 5 basic laws are: 1) \(a^m \cdot a^n = a^{m+n}\), 2) \(\frac{a^m}{a^n} = a^{m-n}\), 3) \((a^m)^n = a^{mn}\), 4) \((ab)^n = a^n b^n\), 5) \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

Q2. What does negative index mean?

A negative index means reciprocal. For example, \(a^{-n} = \frac{1}{a^n}\).

Q3. What does fractional index mean?

A fractional index represents roots. For example, \(a^{\frac{1}{2}} = \sqrt{a}\).

Q4. Why are indices important?

Indices simplify calculations, help express numbers in standard form, and are widely used in algebra, physics, and real-life applications like compound interest.

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Conclusion

The laws of indices are essential in algebra and higher mathematics. By mastering these formulas, you can easily simplify expressions, solve exponential equations, and apply them in real-world problems. Memorize the formulas, practice regularly, and you’ll find indices problems much easier to solve.