lesson 1 part 1 -powers and exponents -second term - prep 1 المنهج الجديد

Introduction

This tutorial focuses on understanding powers and exponents, a fundamental concept in mathematics relevant for students in the second term of prep 1. By mastering these concepts, learners will enhance their problem-solving skills and mathematical reasoning.

Step 1: Understanding Exponents

• Definition: An exponent indicates how many times a number, known as the base, is multiplied by itself.
• Notation: It is written as a small number (the exponent) to the upper right of the base number. For example, ( 2^3 ) means ( 2 \times 2 \times 2 ).

Practical Example:

• Calculate ( 3^2 ):
  • ( 3^2 = 3 \times 3 = 9 )

Step 2: Learning the Power of Zero

• Rule: Any number raised to the power of zero equals one.
• Example:
  • ( 5^0 = 1 )
  • This rule applies to all non-zero numbers.

Step 3: Powers of One

• Rule: Any number raised to the power of one is the number itself.
• Example:
  • ( 7^1 = 7 )

Step 4: Multiplying Powers with the Same Base

• Rule: When multiplying two powers that have the same base, add their exponents.
• Formula: ( a^m \times a^n = a^{m+n} )
• Example:
  • ( 2^3 \times 2^2 = 2^{3+2} = 2^5 = 32 )

Step 5: Dividing Powers with the Same Base

• Rule: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
• Formula: ( a^m \div a^n = a^{m-n} )
• Example:
  • ( 5^4 \div 5^2 = 5^{4-2} = 5^2 = 25 )

Step 6: Powers of a Product

• Rule: When raising a product to a power, raise each factor to the power.
• Formula: ( (ab)^n = a^n \times b^n )
• Example:
  • ( (2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36 )

Step 7: Powers of a Quotient

• Rule: When raising a quotient to a power, raise both the numerator and denominator to that power.
• Formula: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} )
• Example:
  • ( \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} )

Conclusion

Understanding powers and exponents is crucial for solving mathematical problems efficiently. Key rules include how to multiply and divide powers with the same base, as well as the special cases of powers of zero and one. Practice these principles with various examples to reinforce your knowledge. Next steps might include solving practice problems and exploring more complex applications of exponents in algebra.