This website uses cookies to improve your user experience. By continuing to browse, you agree to our use of cookies.
Powers
Definition: A power (or exponent) represents repeated multiplication of a base number.
Example: 2³ means 2 * 2 * 2 = 8
Here, 2 is the base, and 3 is the exponent (or power).
Key Laws of Exponents:
Product Rule: a^m * a^n = a^(m+n)
Example: 2³ * 2⁴ = 2^(3+4) = 2⁷
Quotient Rule: a^m / a^n = a^(m-n)
Example: 3⁵ / 3² = 3^(5-2) = 3³
Power Rule: (a^m)^n = a^(m*n)
Example: (5²)³ = 5^(2*3) = 5⁶
Zero Exponent: a⁰ = 1 (for any non-zero number 'a')
Example: 7⁰ = 1
Negative Exponent: a⁻ⁿ = 1 / aⁿ
Example: 2⁻³ = 1 / 2³ = 1/8
Fractional Exponent: a^(1/n) = ⁿ√a (the nth root of a)
Example: 8^(1/3) = ∛8 = 2
2. Surds
Definition:
Surds are irrational numbers that can be expressed as the root of a rational number.
They include square roots (√), cube roots (∛), fourth roots (∜), and higher-order roots.
Examples: √2, ∛5, ∜16, ⁵√32
Simplifying Surds:
Finding perfect square factors (for square roots):
√18 = √(9 * 2) = √9 * √2 = 3√2
Finding perfect cube factors (for cube roots):
∛54 = ∛(27 * 2) = ∛27 * ∛2 = 3∛2
Generalizing:
The principle applies to all roots. Find factors that are perfect powers of the root index. For example, for fifth roots, look for factors that are perfect fifth powers.
Rationalizing the denominator:
To remove surds from the denominator of a fraction, multiply both the numerator and denominator by a suitable surd.
Example: 1/√2 = (1/√2) * (√2/√2) = √2/2
1 / (√5 - √3)
Multiply numerator and denominator by the conjugate: (√5 + √3)
(1 / (√5 - √3)) * ((√5 + √3) / (√5 + √3))
= (√5 + √3) / ((√5)² - (√3)²)
= (√5 + √3) / (5 - 3)
= (√5 + √3) / 2
Operations with Surds:
Addition and Subtraction:
Can only be performed with like surds (surds with the same number under the root).
Example: 3√2 + 5√2 = 8√2
Multiplication and Division:
Multiply or divide the coefficients and the surds separately.
Example: (2√3) * (4√5) = (2 * 4) * (√3 * √5) = 8√15