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OCR GCSE Maths (Higher) Study Notes

1.3.1 Basic Powers and Roots

Exploring foundational concepts of powers and roots, this section is critical for IGCSE Maths students to grasp the nature of various number types. It simplifies understanding of squares, square roots, cubes, and cube roots, with a focus on recall techniques. The content is designed to be both thorough and succinct, enhancing learning outcomes.

Powers & Indices

Understanding Powers

Powers or indices denote the operation of multiplying a number by itself a specified number of times.

  • 21=22^1 = 2
  • 22=42^2 = 4 (since 2×2=42 \times 2 = 4)
  • 23=82^3 = 8 (as 2×2×2=82 \times 2 \times 2 = 8)
  • 24=162^4 = 16 (as 2×2×2×2=162 \times 2 \times 2 \times 2 = 16)
  • 25=322^5 = 32 (as 2×2×2×2×2=322 \times 2 \times 2 \times 2 \times 2 = 32)
  • 20=12^0 = 1, for any non-zero base.
Powers and Indices

Understanding Roots

Roots perform the inverse operation of powers, identifying the base number from its powered result.

  • Square root: 25=5\sqrt{25} = 5, indicating both 55 and 5-5 are roots.
  • Cube root: 1253=5\sqrt[3]{125} = 5, unique for both positive and negative numbers.

Reciprocals

The reciprocal of a number is what you multiply by to get 1, for example:

  • Reciprocal of 22 is 12\frac{1}{2}.
  • 515^{-1} signifies the reciprocal of 55, and 525^{-2} the reciprocal of 525^2.

Laws of Indices

These laws facilitate the simplification of expressions and computations.

Law of Indices

Worked Examples

Example 1:

Calculate 343^4.

Solution:

34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81

Example 2:

Simplify 49\sqrt{49}.

Solution:

49=7\sqrt{49} = 7

As (7×7=49)(7 \times 7 = 49)

Example 3:

Evaluate (23)3.\left(\frac{2}{3}\right)^{-3}.

Solution:

(23)3=(32)3\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3=3323=278= \frac{3^3}{2^3} = \frac{27}{8}

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