Form Three Chapter One: Indices
Introduction to Indices
- Index Notation/Index Form:
- Base: The number that is multiplied.
- Index/Exponent: Indicates how many times the base is multiplied.
- Example: In (5^2), 5 is the base, 2 is the index.
Key Concepts
1. Repeated Multiplication and Index Form:
- Definition: The value of the index is the number of times the base is multiplied by itself.
- Examples:
- (5^2): Base 5 multiplied 2 times.
- (5^3): Base 5 multiplied 3 times.
- (0.4^2): Base 0.4 multiplied 2 times.
- ((\frac{1}{4})^4): Base (\frac{1}{4}) multiplied 4 times.
2. Convert Numbers to Index Form:
- Method: Repeated multiplication/division to convert.
- Examples:
- ((-p)^7): Negative (p) multiplied 7 times.
- ((1n)^3): 1n multiplied 3 times.
3. Converting to Index Form:
- Methods:
- Repeated Division:
- Example: Convert 16 using base 2 results in (2^4).
- Repeated Multiplication:
- Example: Convert (\frac{32}{3125}) to ((\frac{2}{5})^5).
4. Determine the Value of Numbers in Index Form:
- Methods: Multiplication method or using a calculator.
- Examples:
- (2^5) results in 32.
- ((4/5)^4) results in (\frac{256}{625}).
- Importance of brackets for negative/fractional bases.
Laws of Indices
1. Multiplication of Same Bases:
- Law: (a^m \times a^n = a^{m+n})
- Examples:
2. Division of Same Bases:
- Law: (a^m / a^n = a^{m-n})
- Examples:
3. Raising a Power to a Power:
- Law: ((a^m)^n = a^{m \cdot n})
- Examples:
4. Zero and Negative Indices:
- Laws:
- (a^0 = 1)
- (a^{-n} = \frac{1}{a^n})
- Examples:
- (2^0 = 1)
- (2^{-2} = \frac{1}{2^2})
5. Fractional Indices:
- Law: (a^{\frac{1}{n}} = \sqrt[n]{a})
- Examples:
- (\sqrt{x^2} = x)
- Cube root of (-27 = (-27)^{1/3})
Practice Problems
- Exercises to practice applying these concepts.
Note: Importance of understanding the differences in operation when using brackets with negative bases.
For further details and exercises, refer to the video or contact the teacher for questions!