Algebraic Indices (H)

Higher algebraic indices often combine coefficients, several letter bases, and fractional powers. Simplify the number parts separately, then apply the index laws to each letter base. For fractional powers, remember that the denominator of the power tells you the root.

Common mistake: A power outside brackets applies to every factor inside. For example, \((4x^2)^2 = 16x^4\), not \(4x^4\).

Worked example:
Simplify \(\dfrac{18x^5y^3}{6x^2y}\).
  1. Step 1
    Simplify the coefficient first.
    \[\frac{18}{6} = 3\]
  2. Step 2
    Subtract powers for each matching letter base.
    \[x^{5-2} = x^3\]
    \[y^{3-1} = y^2\]
Answer:
\[\dfrac{18x^5y^3}{6x^2y} = 3x^3y^2\]
Worked example:
Simplify \((64p^6)^{\tfrac{1}{2}}\).
  1. Step 1
    Apply the power to the number part and the algebra part.
    \[64^{\tfrac{1}{2}} = 8\]
    \[(p^6)^{\tfrac{1}{2}} = p^3\]
  2. Step 2
    Multiply the simplified parts.
    \[8p^3\]
Answer:
\[(64p^6)^{\tfrac{1}{2}} = 8p^3\]
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