Higher roots expand our understanding of exponents and radicals. They allow us to work with more complex mathematical expressions, going beyond square roots to cube roots, fourth roots, and beyond. This topic builds on basic exponent rules, introducing new properties and techniques.

Simplifying higher roots, understanding their properties, and performing operations with them are key skills. We'll learn to rationalize denominators with higher roots and explore advanced concepts like fractional exponents and radical equations. These tools are crucial for solving more complex algebraic problems.

Higher Roots

Simplification of higher roots

• Identify the index of the root
  • Small number written above the radical symbol indicates the index
  • $\sqrt[n]{x}$ has an index of $n$ (5th root, cube root)
• Simplify the radicand (expression under the radical symbol)
  • Factor out perfect powers that match the index
    • $\sqrt[3]{8x^3y}=2\sqrt[3]{xy}$ factors out $8=2^3$
  • Combine like terms within the radicand ($3x+4x=7x$)
• Evaluate the root if the radicand is a perfect power matching the index
  • $\sqrt[4]{16}=2$ because $2^4=16$
  • $\sqrt[5]{32}=2$ because $2^5=32$
• Root extraction is the process of finding a number that, when raised to the power of the index, equals the radicand

Properties of higher root expressions

• Product Property: $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$
  • Multiply terms under the radical symbols
  • Keep the same index for the resulting radical
  • $\sqrt[3]{4x}\cdot\sqrt[3]{9y}=\sqrt[3]{36xy}$
• Quotient Property: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
  • Divide terms under the radical symbols
  • Keep the same index for the resulting radical
  • $\frac{\sqrt[4]{16x}}{\sqrt[4]{4y}}=\sqrt[4]{\frac{16x}{4y}}=\sqrt[4]{4\frac{x}{y}}$
• Properties used to simplify expressions with higher roots
• The principal root is the positive root when the index is odd, and the positive or zero root when the index is even

Operations with higher roots

• Like radicals (same index and radicand) can be added or subtracted
  • $2\sqrt[3]{5}+3\sqrt[3]{5}=5\sqrt[3]{5}$
  • $4\sqrt[6]{x}-\sqrt[6]{x}=3\sqrt[6]{x}$
• Unlike radicals cannot be added or subtracted
  • $\sqrt[3]{4}+\sqrt[4]{7}$ cannot be simplified further
  • $5\sqrt{3}-2\sqrt[3]{3}$ cannot be combined
• Simplify each radical term before attempting addition or subtraction
  • $\sqrt[4]{16x}+\sqrt[4]{81x}=\sqrt[4]{x}+3\sqrt[4]{x}=4\sqrt[4]{x}$
  • $\sqrt[3]{27a}-4\sqrt[3]{8a}=3\sqrt[3]{a}-4\cdot2\sqrt[3]{a}=-5\sqrt[3]{a}$

Rationalization of higher root denominators

• Multiply numerator and denominator by a term that eliminates the radical in denominator
  • For a single term denominator: multiply by the radical with an index that makes the denominator a perfect power
    1. $\frac{2}{\sqrt[3]{5}}=\frac{2}{\sqrt[3]{5}}\cdot\frac{\sqrt[3]{25}}{\sqrt[3]{25}}=\frac{2\sqrt[3]{25}}{5}$
    2. $\frac{3}{\sqrt[4]{7}}=\frac{3}{\sqrt[4]{7}}\cdot\frac{\sqrt[4]{343}}{\sqrt[4]{343}}=\frac{3\sqrt[4]{343}}{7}$
  • For a binomial denominator: multiply by the conjugate of the denominator
    • Conjugate obtained by changing the sign between terms
    1. $\frac{1}{\sqrt[4]{3}+\sqrt[4]{2}}=\frac{1}{\sqrt[4]{3}+\sqrt[4]{2}}\cdot\frac{\sqrt[4]{3}-\sqrt[4]{2}}{\sqrt[4]{3}-\sqrt[4]{2}}=\frac{\sqrt[4]{3}-\sqrt[4]{2}}{3^{1/4}-2^{1/4}}$
    2. $\frac{x}{\sqrt[5]{2}-\sqrt[5]{3}}=\frac{x}{\sqrt[5]{2}-\sqrt[5]{3}}\cdot\frac{\sqrt[5]{2}+\sqrt[5]{3}}{\sqrt[5]{2}+\sqrt[5]{3}}=\frac{x(\sqrt[5]{2}+\sqrt[5]{3})}{2^{1/5}-3^{1/5}}$

Advanced Concepts in Higher Roots

• Fractional exponents are an alternative notation for roots, where $\sqrt[n]{x} = x^{\frac{1}{n}}$
• Radical equations are equations where the variable appears under a radical sign
• Real numbers include all rational and irrational numbers, and can be represented on a number line