N2 Powers, roots and standard form.ppt
Powers and Roots Overview
Topic Code: N2
Focus on understanding the concepts related to powers, roots, and standard form.
Powers and Roots
Square Numbers
Definition: A square number results from multiplying a number by itself.
Notation: The square of a number can be written as
n^2.Example:
3 × 3 = 9can also be expressed as3^2 = 9.
Square Numbers: The results of square calculations (0, 1, 4, 9, 16, 25, …).
Square Roots
Definition: The square root is the inverse operation of squaring a number.
Notation: The square root of a number
xis written as√x.Example:
√64 = 8(because8 × 8 = 64).
Product of Square Numbers
Rule: The product of two square numbers is always a square number.
Example:
4 × 25 = 100, because both4and25are squares (2^2and5^2respectively).
Finding Square Roots Using Factors
Use known square factors to compute square roots.
Example:
√400 = √(4 × 100) = √4 × √100 = 2 × 10 = 20.Further Example:
√225 = √(9 × 25) = √9 × √25 = 3 × 5 = 15.
Square Roots of Decimals
Can be found through division of squares.
Example:
√0.09 = √(9 ÷ 100) = √9 ÷ √100 = 3 ÷ 10 = 0.3.Another Example:
√1.44 = √(144 ÷ 100) = √144 ÷ √100 = 12 ÷ 10 = 1.2.
Approximate Square Roots
Rational Numbers: If the square root cannot be expressed exactly, approximate values can be used.
Example:
√2 ≈ 1.414213562(an irrational number).
Estimating Square Roots
Determine bounds for non-square numbers.
Example:
√50 lies between √49 (7) and √64 (8).Approximation:
√50 ≈ 7.07.
Negative Square Roots
Positive and negative roots exist for square numbers.
Example: The square root of
25is5or-5. Notation:±√25indicates both dimensions.
Cube Numbers and Roots
Definition of Cube Numbers
Examples:
1^3 = 1,2^3 = 8,3^3 = 27,4^3 = 64,5^3 = 125.Notation: Written as
n^3, indicating the number multiplied by itself three times.
Cube Roots
Definition: The cube root of a number is the original number that was cubed.
Example: The cube root of
125is5, denoted as∛125 = 5.
Index Notation
What is Index Notation?
Definition: A way to express repeated multiplication succinctly.
Example:
2 × 2 × 2 = 2^3which means two multiplied by itself three times.
Calculating Indices
Calculate values using indices:
Example: Evaluate
6^2 = 36,3^4 = 81,(-5)^3 = -125.
Index Laws
Multiplying Indices
Rule: When multiplying like bases, add the indices.
Example:
a^m × a^n = a^(m+n).
Dividing Indices
Rule: When dividing like bases, subtract the indices.
Example:
a^m ÷ a^n = a^(m-n).
Raising a Power to a Power
Rule: When a base raised to a power is again raised to a power, multiply the indices.
Example:
(x^m)^n = x^(m*n).
Powers of One and Zero
Any number to the power of
1equals the number itself:a^1 = a.Any non-zero number to the power of
0equals1:a^0 = 1.
Negative Indices
Definition of Negative Indices
Rule: A negative exponent indicates a reciprocal.
Example:
a^(-n) = 1/(a^n).
Reciprocals
Simplifying negative indices into fractional form provides the reciprocal of the base.
Fractional Indices
Definition of Fractional Indices
Rule: A fractional exponent signifies roots.
Example:
x^(1/n) = √xandx^(m/n) = n√(x^m).
Surds
Definition of Surds
Numbers that cannot be simplified to remove the square root (for instance,
√3).Note:
√9is not a surd as it equals 3.
Multiplying and Dividing Surds
Basic rules apply when multiplying or dividing surds similar to whole numbers.
Example:
√a × √b = √(ab).
Simplifying Surds
A process of rewriting surds in simplest form,
a√b.Example:
√50 = √(25 × 2) = 5√2.
Standard Form
Definition of Standard Form
Numbers written as
c × 10^n, where1 ≤ c < 10.Examples:
5.97 × 10^24 kgfor the mass of Earth.Used to express very large or small numbers compactly.
Conversion and Calculation in Standard Form
Multiplication: Multiply coefficient parts, add powers.
Division: Divide coefficient parts, subtract powers.
Practical Applications
Used in various fields including astronomy, physics, and finance for ease of calculations with extremely large or small quantities.