Radicals and Rational Exponents

Square Roots: Definition and Evaluation

• Objective: Define and evaluate square roots, focusing on their simplification and relation to radicals.
• Emphasis on Principal Square Root: Only the positive solution is considered.

Properties of Square Roots

• Terminology:
  • Radical: The symbol indicating a root.
  • Radicand: The quantity under the radical.
• When the radicand is a perfect square (e.g., $x^2$), the result is the absolute value of the number $|x|$, ensuring a non-negative result.
• If the square is outside the radical, i.e., $\sqrt{x}^2$, then the result is $x$, given that $x \geq 0$.
• Solving Equations: For $x^2 = a$, the solution is $x = \pm\sqrt{a}$.

Examples of Square Root Simplification

• $\sqrt{121} = 11$ (Principal root)
• $\sqrt{\frac{25}{81}} = \frac{5}{9}$
• $\sqrt{\frac{16}{169}} = \frac{4}{13}$

Properties for Separating Square Roots

• $\sqrt{AB} = \sqrt{A} \cdot \sqrt{B}$ if A and B are non-negative.
• Complex Numbers: Introduction to $i = \sqrt{-1}$.
• $\sqrt{\frac{A}{B}} = \frac{\sqrt{A}}{\sqrt{B}}$ if A and B are non-negative.

Square Root Simplification Examples

• $\sqrt{117} = \sqrt{9 \cdot 13} = \sqrt{9} \cdot \sqrt{13} = 3\sqrt{13}$
• $\sqrt{6} \cdot \sqrt{12} = \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$
• $\sqrt{48x^2} = \sqrt{16 \cdot 3 \cdot x^2} = 4\sqrt{3}|x|$ (Absolute value needed because the value of x is not specified as non-negative).
• $\frac{\sqrt{36}}{\sqrt{3}} = \sqrt{\frac{36}{3}} = \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$
• Simplifying $\sqrt{\frac{25y^3}{9x^2}}$:
  • $= \frac{\sqrt{25y^3}}{\sqrt{9x^2}} = \frac{5\sqrt{y^2 \cdot y}}{3|x|} = \frac{5y\sqrt{y}}{3|x|}$, given $y \geq 0$.

The nth Root

• If A is positive, the nth root exists.
  • Example: Cube root of 8 ($\sqrt[3]{8} = 2$).
• If A is negative:
  • If n is odd, the nth root exists.
  • Example: Cube root of -8 ($\sqrt[3]{-8} = -2$).
  • If n is even, the nth root is not a real number.
  • Example: Fourth root of -16 (DNE).
• If A is zero, the nth root is zero.

Examples: Evaluating nth Roots

• $\sqrt[3]{27} = 3$
• $\sqrt[3]{-64} = -4$
• $\sqrt[4]{16} = 2$ (Principal root)
• $(\sqrt[4]{-3})^4 = \sqrt[4]{81} = 3$
  *Note: If the expression was written as $\sqrt[4]{(-3)^4}$, then simplifying this expression will result in *DNE* because even root doesn't exist with a negative number.*
• $\sqrt[6]{-64}$ – Does Not Exist (DNE).

Simplifying nth Roots

• If n is odd, $\sqrt[n]{a^n} = a$
• If n is even, $\sqrt[n]{a^n} = |a|$ (Principal root).

Properties of nth Roots

• For odd roots, separation is always possible, regardless of the sign.
• For even roots, separation, $\sqrt[n]{AB} = \sqrt[n]{A} \cdot \sqrt[n]{B}$, and division, $\sqrt[n]{\frac{A}{B}} = \frac{\sqrt[n]{A}}{\sqrt[n]{B}}$, requires A and B to be positive.
• Nested Radicals: $\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}$

Examples of Simplifying nth Roots

• $\sqrt[4 \cdot 3]{7} = \sqrt[12]{7}$
• $\sqrt[3]{135} = \sqrt[3]{27 \cdot 5} = 3\sqrt[3]{5}$
• $\frac{\sqrt[3]{5}}{\sqrt[3]{64}} = \frac{\sqrt[3]{5}}{4}$
• $\sqrt[4]{162a^4} = \sqrt[4]{81 \cdot 2 \cdot a^4} = 3 \sqrt[4]{2} |a|$

Addition and Subtraction of Radicals

• Combine like terms only.
• Example: $2\sqrt{45} + 7\sqrt{20}$
  • $= 2\sqrt{9 \cdot 5} + 7\sqrt{4 \cdot 5}$
  • $= 2(3\sqrt{5}) + 7(2\sqrt{5})$
  • $= 6\sqrt{5} + 14\sqrt{5} = 20\sqrt{5}$
• Example: $5\sqrt[3]{8x} - 3\sqrt[3]{27x}$
  • $= 5\sqrt[3]{8 \cdot x} - 3\sqrt[3]{27 \cdot x}$
  • $= 5(2\sqrt[3]{x}) - 3(3\sqrt[3]{x})$
  • $= 10\sqrt[3]{10x} - 9\sqrt[3]{10x} = \sqrt[3]{10x}$

Rational Exponents and Radicals

• Definition: $a^{\frac{1}{n}} = \sqrt[n]{a}$
• Rules for Rational Exponents follow nth root rules.
• $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

Examples: Rational Exponents

• $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
• $(-16)^{\frac{5}{2}}$ – Does Not Exist (DNE).
• $(\sqrt{100})^{-3} = 10^{-3} = \frac{1}{1000}$
• $(-64)^{\frac{2}{6}}$ – Cannot be solved, DNE.