Introduction to Index Laws
Index laws (or exponent rules) are a set of rules that simplify expressions involving powers. Understanding these laws is crucial when dealing with algebraic expressions, especially those involving integer indices.
Basic Index Laws
- Law 1: Product of Powers
When multiplying two expressions with the same base, you add the indices:
am × an = am+n - Law 2: Quotient of Powers
When dividing two expressions with the same base, you subtract the indices:
am ÷ an = am-n - Law 3: Power of a Power
When raising a power to another power, you multiply the indices:
(am)n = am×n - Law 4: Power of a Product
When raising a product to a power, you apply the exponent to each factor:
(ab)n = an × bn - Law 5: Power of a Quotient
When raising a quotient to a power, you apply the exponent to both the numerator and the denominator:
(a/b)n = an ÷ bn - Law 6: Zero Exponent Law
Any non-zero base raised to the power of zero equals one:
a0 = 1(for a ≠ 0) - Law 7: Negative Exponent Law
A negative exponent represents the reciprocal of the base raised to the positive exponent:
a-n = 1/an(for a ≠ 0)
Applying Index Laws with Examples
Example 1: Simplifying Expressions
Simplify the expression: 23 × 22
- Identify the bases: Here, the base is 2.
- Apply the Product of Powers Law:
23 × 22 = 23+2 = 25 - Calculate:
25 = 32
Example 2: Handling Negative Exponents
Simplify the expression: 3-2
- Apply the Negative Exponent Law:
3-2 = 1/32 - Calculate:
1/32 = 1/9
Practice Problems
Try applying these laws on your own:
- Simplify:
54 ÷ 52 - Simplify:
(x3)2 - Simplify:
(2 × 3)3
Conclusion
By mastering index laws, you will find that simplifying and manipulating algebraic expressions becomes easier. Keep practicing with various problems to become proficient in using these laws effectively!