BLE & SEE Maths: Laws of Indices Formulas with Examples

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BLE & SEE Maths: Laws of Indices Formulas with Examples

Hello!! Dear students, teachers and parents, Welcome back to Bidhya Mandir, your smart e-Learning Platform of Nepal for guide to academic success! In both the Basic Level Examination (BLE – Class 8) and the Secondary Education Examination (SEE – Class 10), Algebra is very important chapter. One of the easiest ways to secure full marks in this section is by knowing the Laws of Indices (Exponents). In this post, We are very excited to share some law of indices with examples in a simple and easy specially for SEE and BLE students. This is titled as “BLE & SEE Maths: Laws of Indices Formulas with Examples”

Whether you are simplifying long algebraic fractions or solving exponential equations, these core formulas are your ultimate weapons. Let’s know them down with simple definitions and examples.

What are Indices?

Firstly, we should understand the anatomy of the index or exponent before jumping into the formulas, let’s understand the anatomy of an index. In the given algebraic expression,

an a^n

  • a is called the Base (the number that is being multiplied).
  • n is called the Index, Exponent or Power (the number of times the base is multiplied by itself).

The Laws of Indices Formula Chart

Here is the best summary table of all the laws of indices you need to memorize for your exams:

Name of LawsMathematical FormulaDescription
Product Lawam×an=am+na^m \times a^n = a^{m+n}If the bases are the same, add the powers during multiplication.
Quotient Lawam÷an=aman=am−na^m \div a^n = \frac{a^m}{a^n} = a^{m-n}If the bases are the same, subtract the powers during division.
Power Law(am)n=am×n(a^m)^n = a^{m \times n}A power raised to another power means you multiply the powers.
Zero Index Lawa0=1(a≠0)a^0 = 1 \quad (a \neq 0)Any non-zero base raised to the power of zero is always equal to 1.
Negative Index Lawa−m=1amor(ab)−m=(ba)ma^{-m} = \frac{1}{a^m} or \left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^mTo make a negative power positive, take the reciprocal of the base.
Fractional Index Lawamn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}The denominator of a fractional power becomes the root ()(\sqrt{}).
Product Power Law(ab)n=an⋅bn(ab)^n = a^n \cdot b^nDistribute the power to every base inside the bracket.

Let’s Learn Formulas with Examples

Let’s look at how these formulas work in actual exam-style problems.

1. The Product Law

Formula: am×an=am+na^m \times a^n = a^{m+n}

  • Example: Simplify: x5×x3x^5 \times x^3
  • Solution: x5+3=x8x^{5+3} = x^8

2. The Quotient Law

Formula: aman=am−n \frac{a^m}{a^n} = a^{m-n}

  • Example: Simplify: y7y3 \frac{y^7}{y^3}
  • Solution: y7−3=y4y^{7-3} = y^4

3. The Power Law

Formula: (am)n=am⋅n(a^m)^n = a^{m \cdot n}

  • Example: Simplify: (23)2(2^3)^2
  • Solution: 23×2=26=642^{3 \times 2} = 2^6 = 64

4. The Zero Index Law

Formula: a0=1a^0 = 1

  • Example: Evaluate: (5x2y3)0(5x^2y^3)^0
  • Solution: Since the entire expression is raised to the power of 0, the answer is simply 1.

5. The Negative Index Law

Formula: a−m=1ama^{-m} = \frac{1}{a^m}

  • Example: Evaluate 3−23^{-2}
  • Solution: 132=19\frac{1}{3^2} = \frac{1}{9}
  • Bonus BLE/SEE Trick: If you have (23)−3\left(\frac{2}{3}\right)^{-3}, just flip the fraction to make it positive: (32)3=278\left(\frac{3}{2}\right)^3 = \frac{27}{8}.

6. The Root/Fractional Index Law

Formula: a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}

  • Example: Evaluate: 271327^{\frac{1}{3}}
  • Solution: This means the cube root of 27→273=333=3.27 \rightarrow \sqrt[3]{27} = \sqrt[3]{3^3} = 3.

Important Tips for BLE & SEE Board Exams

Check the Bases First: You can only add or subtract powers if the bases are exactly the same. You cannot simplify x2×y3x^2 \times y^3 into one power!

Break Down Composite Numbers: If you see numbers like 4, 8, 9, or 27 in an indices problem, break them down into prime bases first (e.g., turn 9x9^x into (32)x=32x).(3^2)^x = 3^{2x}).

Don’t Forget the Signs: Be very careful with minus signs when applying the Quotient Law, especially if the denominator already has a negative power (e.g., x3x−2=x3−(−2)=x3+2=x5).\frac{x^3}{x^{-2}} = x^{3 – (-2)} = x^{3+2} = x^5).

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