The Laws of Indices (Powers)

Indices (or powers) are a compact way to write repeated multiplication. For a non-zero number a and integers m, n, the expression an means that a is multiplied by itself n times.

1. Product Law (Same Base)

Rule: When you multiply powers with the same base, you add the indices:

am × an = am + n

Example:

23 × 24 = 23 + 4 = 27

2. Quotient Law (Same Base)

Rule: When you divide powers with the same base, you subtract the indices:

am / an = am − n,   a ≠ 0

Example:

57 / 53 = 57 − 3 = 54

3. Power of a Power

Rule: When you raise a power to another power, you multiply the indices:

(am)n = amn

Example:

(32)4 = 32 × 4 = 38

4. Power of a Product

Rule: When you raise a product to a power, each factor is raised to that power:

(ab)n = anbn

Example:

(2 × 5)3 = 23 × 53 = 8 × 125 = 1000

5. Power of a Quotient

Rule: When you raise a fraction to a power, both numerator and denominator are raised to that power:

(a / b)n = an / bn,   b ≠ 0

Example:

(3 / 4)2 = 32 / 42 = 9 / 16

6. Zero Index

Rule: Any non-zero number raised to the power zero is equal to 1:

a0 = 1,   a ≠ 0

Example:

70 = 1,   1000 = 1,   (−3)0 = 1

(We do not define 00 here – it is left undefined in most contexts.)

7. Negative Indices

Rule: A negative index moves the factor to the denominator and makes the index positive:

a−n = 1 / an,   a ≠ 0

Examples:

  • 2−3 = 1 / 23 = 1 / 8
  • 5−1 = 1 / 5

8. Fractional Indices (Roots)

Rule: A fractional index represents a root. For a positive number a:

a1/n means “the n-th root of a”, with a > 0.

Example:

91/2 = √9 = 3,   81/3 = the cube root of 8 = 2

9. General Fractional Indices

Rule: For a rational index p/q with integers p, q and q > 0:

ap/q = (a1/q)p = (the q-th root of a)p,   a > 0

Examples:

  • 163/4 = (161/4)3 = 23 = 8
  • 272/3 = (271/3)2 = 32 = 9

Summary of the Laws of Indices

  • Product: am × an = am + n
  • Quotient: am / an = am − n,   a ≠ 0
  • Power of a power: (am)n = amn
  • Power of a product: (ab)n = anbn
  • Power of a quotient: (a / b)n = an / bn,   b ≠ 0
  • Zero index: a0 = 1,   a ≠ 0
  • Negative index: a−n = 1 / an,   a ≠ 0
  • Fractional index: ap/q = (a1/q)p,   a > 0

Popular posts from this blog

A Geometric Way to Visualise sin(x + y) and cos(x + y)

Image
The angle–addition identities for sine and cosine often appear as algebraic formulas, but they can also be understood by combining two right triangles in a simple geometric construction. The calculations for the side lengths follow directly from the definitions of sine and cosine. sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y − sin x sin y Start with a right triangle of angle y and hypotenuse 1. From basic trigonometry, its horizontal and vertical sides are: cos y and sin y. Next, attach a second right triangle with angle x . Its hypotenuse is the side of length cos y from the first triangle, so its adjacent and opposite sides become: adjacent = cos x · cos y opposite = sin x · cos y Likewise, if the first triangle's vertical side sin y is used as a hypotenuse in a similar way, it contributes: adjacent = cos x · sin y opposite = sin x · sin y When the horizontal components are combined, they give the expression for cos(x + y...
Read more

The Method of Differences — A Clean Proof of the Sum of Cubes

The Method of Differences — A Clean Proof of the Sum of Cubes The method of differences is a remarkably elegant tool for evaluating finite sums. When each term of a series can be written in the form f(r+1) − f(r) , the sum “collapses” — all interior terms cancel, leaving only a boundary expression. This behaviour is called a telescoping sum . 1) Telescoping Sums Assume the general term u r can be written as: u r = f(r+1) − f(r). Then the finite sum from r = 1 to r = n becomes: Σ u r = Σ ( f(r+1) − f(r) ). To see what happens, write out a few terms: u₁ = f(2) − f(1) u₂ = f(3) − f(2) u₃ = f(4) − f(3) ⋮ uₙ = f(n+1) − f(n) When these are added, everything cancels except the first and last pieces: Σ u r = f(n+1) − f(1). This is the essence of the method: interior structure disappears, leaving just the difference between the final and initial states. 2) A Classic Application — The Sum of Cubes We will use this technique to prove the well-known ...
Read more

2×2 Orthogonal Matrix Mastery — A Generalised Construction

Image
2×2 Orthogonal Matrix Mastery — A Generalised Construction Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions: They are perpendicular (their dot product is zero); They have unit length (their magnitude is one). This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix. 1. Begin with two perpendicular vectors Let the first vector be: (a, b) A perpendicular vector can be chosen as: (−b, a) Their dot products confirm orthogonality: (a, b) · (−b, a) = −ab + ab = 0 (a, −b) · (b, a) = ab − ab = 0 2. Compute their shared magnitude Both vectors have the same length: |(a, b)| = √(a² + b²) We can therefore normalise each one by dividing by √(a² + b²). 3. Form the matrix using the normalised vectors Place the two normalised vectors...
Read more