0580

Algebraic Roots and Indices

Algebra and Sequences

Matching the bases

  • Where an unknown appears in the index, rewrite both sides using the same base, then equate the indices
  • This works because a single base raised to two powers is equal only when the powers match
  • Recognising powers of 2, 3 and 5 is what makes the rewriting possible
Exam tip

A power outside a bracket hits the coefficient too: (3x²)³ is 27x⁶, not 3x⁶. Index rules only work on a shared base, so check that before adding or subtracting powers. And a negative index gives a reciprocal, never a negative answer.

Worked example

Solving an index equation

Solve 2^(x + 3) = 8ˣ.

Solution:

  • Write 8 as a power of 2: 8 = 2³
  • The right-hand side becomes (2³)ˣ, which is 2^(3x)
  • Both sides now share base 2, so the indices are equal: x + 3 = 3x
  • Rearranging gives 3 = 2x, so x = 1.5
  • Where the unknown side is the smaller power, the answer is often a fraction
    • 27ˣ = 3 becomes 3^(3x) = 3¹, so 3x = 1 and x = 1/3
  • Logarithms are not required by this specification, so every such equation can be solved by matching bases