1.5 Laws of Exponents for Real Numbers

Description

Quick Overview

This section introduces the laws of exponents for real numbers, covering operations with positive rational exponents and their extensions.

Standard

In this section, students explore the laws of exponents applied to real numbers, including negative and rational exponents. It emphasizes their definitions, properties, and application through examples and exercises for mastering mathematical operations involving exponents.

Detailed

Laws of Exponents for Real Numbers

This section details the laws of exponents and how to apply them to real numbers. Key concepts of exponents are first reviewed, including the definitions and basic laws learned in earlier classes. The laws established include:

  1. Product of Powers: When multiplying two powers with the same base, add the exponents: a^m * a^n = a^{m+n}.
  2. Power of a Power: When raising a power to another power, multiply the exponents: (a^m)^n = a^{mn}.
  3. Quotient of Powers: When dividing two powers with the same base, subtract the exponents: a^m / a^n = a^{m-n}.
  4. Power of a Product: When raising a product to a power, distribute the exponent to both factors: (ab)^m = a^m * b^m.

The section also addresses the concept of zero and negative exponents, introducing definitions that allow for rational numbers to have fractional exponents: n√a = a^{1/n}.

Finally, examples provide guidance in simplifying expressions with rational and negative exponents. Students are encouraged to utilize these laws to solve problems involving exponents effectively.

Example 21: Simplify

(i) \( \frac{2^4 \cdot 2^3}{2^6} \)

(ii) \( \frac{3^7}{3^2 \cdot 3^3} \)

(iii) \( \frac{5^9}{(5^3)^2} \)

(iv) \( \frac{7^4 \cdot 7^5}{7^6} \)

Solution :

(i) \( \frac{2^4 \cdot 2^3}{2^6} = \frac{2^{4+3}}{2^6} = 2^{7-6} = 2^{1} = 2 \)

(ii) \( \frac{3^7}{3^2 \cdot 3^3} = \frac{3^7}{3^{2+3}} = 3^{7-5} = 3^{2} = 9 \)

(iii) \( \frac{5^9}{(5^3)^2} = \frac{5^9}{5^{3 \cdot 2}} = 5^{9-6} = 5^{3} = 125 \)

(iv) \( \frac{7^4 \cdot 7^5}{7^6} = \frac{7^{4+5}}{7^{6}} = 7^{9-6} = 7^{3} = 343 \)

Key Concepts

  • Product of Powers: When multiplying, add the exponents.

  • Power of a Power: When raising to a power, multiply the exponents.

  • Quotient of Powers: When dividing, subtract the exponents.

  • Rational Exponents: Represent roots using fractional exponents.

Memory Aids

🎡 Rhymes Time

  • To multiply, simply add, it’s really not so bad!

πŸ“– Fascinating Stories

  • Imagine a baker adding two cakes (exponents) together when stacking them up.

🧠 Other Memory Gems

  • N.E.R.D - Negative Exponents Require Division!

🎯 Super Acronyms

P.A.S.T. - Product Adds Same base's exponents Together!

Examples

  • Example of product of powers: 2^3 * 2^4 = 2^{3+4} = 2^7.

  • Example of power of a power: (3^2)^3 = 3^{2*3} = 3^6.

Glossary of Terms

  • Term: Exponent

    Definition:

    A mathematical notation indicating the number of times to multiply a quantity by itself, e.g., in a^n, n is the exponent.

  • Term: Base

    Definition:

    The number that is being raised to a power in exponential notation.

  • Term: Rational Exponent

    Definition:

    An exponent that can be expressed as a fraction m/n where m and n are integers.