Algebraic Roots And Indices

## Algebraic Roots & Indices ### Laws of Indices - **Multiplication**: \(a^m \times a^n = a^{m+n}\) - **Division**: \(a^m \div a^n = a^{m-n}\) - **Power of a power**: \((a^m)^n = a^{mn}\) - **Power of a product**: \((ab)^n = a^n b^n\) - **Power of a fraction**: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) - **Zero exponent**: \(a^0 = 1\) - **Negative exponent**: \(a^{-n} = \frac{1}{a^n}\) - **Fractional exponent**: \(a^{\frac{1}{n}} = \sqrt[n]{a}\), \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\) - **Negative fractional**: \(a^{-\frac{1}{n}} = \frac{1}{\sqrt[n]{a}}\) ### Simplifying Expressions - Work out number and algebra parts separately. - Example: \((3x^7) \times (6x^4) = 18x^{11}\) - Example: \(\frac{6x^7}{3x^4} = 2x^3\) - Example: \((3x^7)^2 = 9x^{14}\) ### Solving Equations with Unknown Powers - If both sides have the same base, equate the exponents. - Example: \(4^{3x} = 4^9 \Rightarrow 3x = 9 \Rightarrow x = 3\) - Simplify first if needed: \(3^{2x} \times 3^4 = 3^{18} \Rightarrow 3^{2x+4} = 3^{18} \Rightarrow 2x+4=18 \Rightarrow x=7\) ### Worked Examples 1. Simplify \((u^5)^5 = u^{25}\) 2. If \(q^x = \frac{q^2 \times q^5}{q^{10}}\), find \(x\). Simplify numerator: \(q^7\), then \(\frac{q^7}{q^{10}} = q^{-3}\), so \(x = -3\). 3. Rewrite \(\frac{1}{\sqrt[3]{x^4}}\) as \(x^n\). \(\sqrt[3]{x^4} = x^{4/3}\), so \(\frac{1}{x^{4/3}} = x^{-4/3}\). 4. Find constants \(m\) and \(a\) given \((ax^6)^{1/m} = 8x^3\). Expand: \(a^{1/m} x^{6/m} = 8x^3\). Equate \(x\) powers: \(6/m = 3 \Rightarrow m=2\). Then \(a^{1/2}=8 \Rightarrow a=64\). ### Key Tips - Always simplify using index laws before solving. - For negative fractional powers, take reciprocal first. - When equating, match coefficients and powers separately.