Algebraic indices extend the laws of indices from numbers to expressions involving letters. The same rules apply.
Laws of indices: $$a^m \times a^n = a^{m+n}$$ $$a^m \div a^n = a^{m-n}$$ $$(a^m)^n = a^{mn}$$ $$a^0 = 1$$ $$a^{-n} = \frac{1}{a^n}$$ $$a^{\frac{1}{n}} = \sqrt[n]{a}$$ $$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$
Examples: $$\frac{x^5 \times x^3}{x^4} = \frac{x^8}{x^4} = x^4$$ $$(4x^3)^2 = 16x^6$$
Simplifying expressions: collect coefficients separately from powers. $$\frac{6a^4b^3}{3a^2b} = 2a^2b^2$$
Common error: multiplying indices when bases are multiplied (e.g. $x^2 \times x^3 = x^6$ — wrong, it's $x^5$). Also: $(ab)^2 = a^2b^2$, not $ab^2$.
Indices Fractional & negative indices Standard form
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