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Understanding Power and Index

In simple terms, power and index involve expressing a number (called the base) raised to a certain value (the exponent or index). This results in a new number, which represents the base multiplied by itself multiple times. The exponent indicates how many times the base is multiplied by itself.

Basic Notation

  • Base: The number being raised to a power.
  • Exponent/Index: The value that indicates how many times the base is multiplied by itself.
  • Result/Value: The outcome of the exponentiation, also known as the power.

Example: In the expression 23, 2 is the base and 3 is the exponent. The result is 2 * 2 * 2 = 8.

Solving Power and Index Problems

Example 1: Simplifying Expressions
(i) Simplify 52 * 53.
Sol:

  • Apply the rule of exponents: When multiplying numbers with the same base and different exponents, add the exponents.
  • 52 * 53 = 5(2+3) = 55 = 5 * 5 * 5 * 5 * 5 = 3125.

Example 2: Negative Exponents
(i) Simplify 2-3.
Sol:

  • A negative exponent indicates the reciprocal of the base raised to the positive exponent.
  • 2-3 = 1 / 23 = 1 / (2 * 2 * 2) = 1/8.

Example 3: Fractional Exponents
(i) Evaluate 16(1/2).
Sol:

  • A fractional exponent represents the root of the base. A square root is the same as raising to the power of 1/2.
  • 16(1/2) = √16 = 4.

Strategies for Solving Power and Index Problems

  • Familiarise with Laws of Exponents: Understand the laws of exponents, including rules for multiplication, division, powers of powers and negative exponents.
  • Practice Mental Calculation: Memorise common powers of numbers (like 20, 21, 22, etc.) to simplify calculations.
  • Stay Organised: When dealing with complex expressions, organise your work by applying the rules of exponents step by step.
  • Use Calculator Wisely: For calculations involving large numbers, decimals or fractional exponents, use calculators effectively.
  • Solve Backwards: In some problems, you might need to find the exponent given the result. Use the reverse process of taking logarithms to solve for the exponent.

Conclusion

Mastering Power and Index concepts is essential for enhancing your mathematical problem-solving skills. Understanding how to manipulate numbers with exponents can significantly simplify calculations and help you tackle complex expressions efficiently. By familiarising yourself with the laws of exponents, practicing mental calculation and employing strategic problem-solving techniques, you’ll be well-prepared to tackle Power and Index problems confidently in the CLAT and other competitive exams.

The document Overview: Power and Index | Quantitative Techniques for CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
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