Algebraic Indices

Algebraic indices show repeated multiplication, such as \(x^3 = x \times x \times x\). When multiplying terms with the same base, add the powers. When a power is raised to another power, multiply the powers.

Common mistake: Only use the index laws when the bases match. \(a^2 \times b^3\) cannot be simplified to \(ab^5\), because the bases are different.

Worked example:
Simplify \(x^3 \times x^2\).
  1. The base is the same, so add the powers.
    \[x^3 \times x^2 = x^{3+2}\]
    \[x^{3+2} = x^5\]
Answer:
\[x^5\]
Worked example:
Simplify \(\dfrac{12k^7}{4k^2}\).
  1. Divide the coefficients, then subtract powers because the base is the same.
    \[12 \div 4 = 3\]
    \[k^7 \div k^2 = k^{7-2} = k^5\]
Answer:
\[3k^5\]
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