OCR GCSE Maths: Powers and Roots

OCR GCSE Maths: Powers and Roots

Understanding Powers

• Definition: A power represents repeated multiplication of a base number by itself.
• Notation: We use exponents (small superscripts) to indicate the power. For example, 2^3 represents 2 multiplied by itself three times (2 x 2 x 2 = 8).
  • 2 is the base
  • 3 is the exponent or index
  • The entire expression 2^3 is read as "2 to the power of 3" or "2 cubed."

Positive Integer Powers

• Calculation: To calculate a positive integer power, simply multiply the base by itself the number of times indicated by the exponent.
  • Example: 5^2 = 5 x 5 = 25

Negative Integer Powers

• Definition: A negative exponent indicates the reciprocal of the corresponding positive power.
• Calculation: To calculate a negative integer power, take the reciprocal of the base raised to the positive value of the exponent.
  • Example: 2^-3 = 1 / 2^3 = 1 / (2 x 2 x 2) = 1/8

Roots

• Definition: A root of a number is a value that, when multiplied by itself a specific number of times, equals the original number.
• Notation: The radical symbol (?) represents a root. The number above the radical symbol (index) indicates the type of root to be found.
  • ?x represents the square root of x (the value that, when multiplied by itself, equals x).
  • ³?x represents the cube root of x (the value that, when multiplied by itself three times, equals x).

Calculating Exact Roots

• Square Roots: You should be able to recognize the perfect squares (numbers that result from squaring an integer) up to 12² (144).
  • Example: ?64 = 8 because 8² = 64
• Cube Roots: Familiarize yourself with the cubes of small integers (1³, 2³, 3³, 4³, 5³).
  • Example: ³?27 = 3 because 3³ = 27

Simple Powers of 2, 3, 4, and 5

• Memorization: Memorizing the powers of 2, 3, 4, and 5 up to their fifth power will be helpful in solving problems.
• Example:
  • 2² = 4, 2³ = 8, 2? = 16, 2? = 32
  • 3² = 9, 3³ = 27, 3? = 81, 3? = 243

Applying Powers and Roots in Numerical Contexts

• Area and Volume: Powers are used to calculate areas (squares) and volumes (cubes).
  • Example: The area of a square with side length s is s².
• Scientific Notation: Powers of 10 are used to represent very large or very small numbers in scientific notation.
• Fractions: Powers can be used to simplify fractions with exponents.
  • Example: (2/3)^2 = (2/3) x (2/3) = 4/9

Practice and Application

• Practice calculating powers and roots using different integer indices.
• Solve problems involving powers and roots in different numerical contexts.
• Apply the concepts of powers and roots to real-world scenarios (e.g., calculating areas and volumes).