Year 10 Powers And Surds Flashcards
Surd
Root of a number
Cannot be simplified into a rational number
Laws of Surds
√a x √b = √ab
√a / √b = √(a / b)
Form a√b
b must be a surd, root that cannot be simplified, so not a square number.
√8 = √4 √2 = 2 √2
Adding Surds
Like addition of unknowns in algebra
√3+ √3 2 √3
√4 + √4 + √4 = 3 √4
3 √12 + √27 = 3 √4 √3 + √9 √3 = 6 √3 + 3 √3 = 9 √3
Rational number
Number can be written as a fraction, decimal form has either a terminating or recurring decimal, such as 12.4
Irrational number
Numbers cannot be written as a fraction: such as √2
Dividing Surds
6 / √3
= 6 √3 / 3
= 2 √3
(Multiply top and bottom of the expression by √3 that is the denominator)
If the denominator is more complex:
3 / √6 - 2: find the conjugate (denominator but the + or - signs reversed)
= 3 (√6 + 2) / (√6 - 2)(√6 + 2): multiply both bottom and top of the expression by the conjugate.
= 3 √6 + 6 / 1
= 3 √6 + 6
When multiplying indices
Add up exponents:
x^3 x x^2 = x^5
When dividing indices
Substrat exponents x^5 / x^3 = x^2
Only works for real numbers / integers
Indices in brackets
Multiply exponents
(x^3)^4 = x^12
Works for all real numbers / integers
Bases to fractional powers
³√x
= x¹/³
the denominator becomes the root.
Only applies if the numerator is 1
9³/²
= (9³)¹/²
= ²√9³
= 27
In the fractional power:
denominator becomes how many times to root the number
Numerator becomes the power of the base
Rationalising the denominator
Making the denominator a rational number
1 + √5 / 3 + √5
= (1 + √5)(3 - √5) / (3 + √5)(3 - √5): multiplying the top and bottom by the conjugate which is (3 - √5)
= 3 - √5 / 9 - √25
= 3 - √5 / 9 - 5
= 3 √5 / 4
Negative indices
6⁻¹ = 1 / 6
6 ⁻² = 1 / 6² = 1 / 36
Changing bases
Example:
4ˣ = 2¹⁰
2ˣ 2ˣ = 2¹⁰: make all the bases in the equation the same number
2²ˣ = 2¹⁰: follows (ab) ⁿ = a ⁿb ⁿ
2²⁽⁵⁾ = 2¹⁰
So x = 5
Calculate the value and write the answer in surd form:
³⁶√60 / ⁹√12
⁴√5
Divide powers by powers
Divide bases by bases
