📌 What is a Radical?

A radical expression contains a root symbol \( \sqrt[n]{a} \) where:

• \( n \) is the index (degree of the root)
• \( a \) is the radicand (number under the root)
• \( \sqrt[2]{a} = \sqrt{a} \) is a square root (index 2 is usually omitted)
• \( \sqrt[3]{a} \) is a cube root

Perfect Squares:

\( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \), \( 5^2 = 25 \), \( 6^2 = 36 \),
\( 7^2 = 49 \), \( 8^2 = 64 \), \( 9^2 = 81 \), \( 10^2 = 100 \), \( 11^2 = 121 \), \( 12^2 = 144 \)

Perfect Cubes:

\( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \), \( 5^3 = 125 \),
\( 6^3 = 216 \), \( 7^3 = 343 \), \( 8^3 = 512 \), \( 9^3 = 729 \), \( 10^3 = 1000 \)

Roots of Rational Numbers:

For a fraction, take the root of numerator and denominator separately:

\( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) where \( b \neq 0 \)

Definition:

The nth root of \(a\) is the number that, when raised to the \(n\)th power, equals \(a\).

If \( b^n = a \), then \( b = \sqrt[n]{a} \)

📝 Examples:

• \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \)
• \( \sqrt[4]{16} = 2 \) because \( 2^4 = 16 \)
• \( \sqrt[5]{32} = 2 \) because \( 2^5 = 32 \)
• \( \sqrt[3]{-27} = -3 \) because \( (-3)^3 = -27 \)

Product Property:

\( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab} \)

For \( n \geq 2 \) and real numbers \(a\) and \(b\)

Quotient Property:

\( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \)

For \( n \geq 2 \), \( b \neq 0 \), and real numbers \(a\) and \(b\)

Power Property:

\( (\sqrt[n]{a})^n = a \) and \( \sqrt[n]{a^n} = |a| \) (for even \(n\))

Simplified Form Conditions:

A radical expression is in simplified form when:

1. No radicands have perfect nth powers as factors (other than 1)
2. No radicands contain fractions
3. No radicals appear in the denominator

Steps to Simplify:

1. Find the largest perfect power factor in the radicand
2. Rewrite the radicand as a product of factors
3. Apply the product property to separate radicals
4. Simplify the perfect power roots

📝 Examples (Square Roots):

\( \sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \)
\( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \)
\( \sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2} \)
\( \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2} \)

📝 Examples (Cube Roots):

\( \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = \sqrt[3]{27} \cdot \sqrt[3]{2} = 3\sqrt[3]{2} \)
\( \sqrt[3]{128} = \sqrt[3]{64 \cdot 2} = \sqrt[3]{64} \cdot \sqrt[3]{2} = 4\sqrt[3]{2} \)
\( \sqrt[3]{250} = \sqrt[3]{125 \cdot 2} = 5\sqrt[3]{2} \)

Rules for Variables:

When simplifying radicals with variables, factor out the largest perfect power:

📝 Examples with Variables:

\( \sqrt{x^5} = \sqrt{x^4 \cdot x} = x^2\sqrt{x} \)
\( \sqrt{18x^3} = \sqrt{9 \cdot 2 \cdot x^2 \cdot x} = 3x\sqrt{2x} \)
\( \sqrt{50a^7b^4} = \sqrt{25 \cdot 2 \cdot a^6 \cdot a \cdot b^4} = 5a^3b^2\sqrt{2a} \)
\( \sqrt[3]{24x^5y^6} = \sqrt[3]{8 \cdot 3 \cdot x^3 \cdot x^2 \cdot y^6} = 2xy^2\sqrt[3]{3x^2} \)

Basic Rule:

To multiply radicals with the same index:

1. Multiply the coefficients (numbers outside)
2. Multiply the radicands (numbers inside)
3. Simplify if possible

📝 Examples:

\( \sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6 \)
\( 2\sqrt{5} \cdot 3\sqrt{7} = 6\sqrt{35} \)
\( 4\sqrt{6} \cdot 2\sqrt{3} = 8\sqrt{18} = 8\sqrt{9 \cdot 2} = 24\sqrt{2} \)
\( \sqrt{2x} \cdot \sqrt{8x^3} = \sqrt{16x^4} = 4x^2 \)

Like Radicals:

Like radicals have the same index and the same radicand. Only like radicals can be combined.

\( a\sqrt[n]{c} + b\sqrt[n]{c} = (a+b)\sqrt[n]{c} \)

Add or subtract the coefficients, keep the radical part the same

📝 Examples:

\( 3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5} \)
\( 8\sqrt{3} - 2\sqrt{3} = 6\sqrt{3} \)
\( 5\sqrt[3]{7} + 2\sqrt[3]{7} - 3\sqrt[3]{7} = 4\sqrt[3]{7} \)
\( 4\sqrt{x} + 9\sqrt{x} = 13\sqrt{x} \)

⚠️ Cannot Be Combined:

\( \sqrt{2} + \sqrt{3} \) (different radicands)
\( \sqrt{5} + \sqrt[3]{5} \) (different indices)

📝 Simplify First, Then Add/Subtract:

Sometimes radicals must be simplified before they can be combined:

\( \sqrt{48} + \sqrt{27} \)
\( = \sqrt{16 \cdot 3} + \sqrt{9 \cdot 3} \)
\( = 4\sqrt{3} + 3\sqrt{3} \)
\( = 7\sqrt{3} \)

\( \sqrt{50} - \sqrt{18} + \sqrt{32} \)
\( = 5\sqrt{2} - 3\sqrt{2} + 4\sqrt{2} \)
\( = 6\sqrt{2} \)

Quotient Property:

\( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) where \( b > 0 \)

Rationalizing the Denominator:

Rationalizing means eliminating radicals from the denominator.

Method: Multiply numerator and denominator by the radical in the denominator

📝 Examples:

\( \frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \)

\( \frac{8}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \)

\( \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2} \)

\( \frac{12}{\sqrt{6}} = \frac{12\sqrt{6}}{6} = 2\sqrt{6} \)

Distributive Property:

\( a(b + c) = ab + ac \)

This property works the same way with radicals

📝 Examples:

\( \sqrt{3}(5 + \sqrt{2}) = 5\sqrt{3} + \sqrt{6} \)

\( 2\sqrt{5}(\sqrt{3} + 4) = 2\sqrt{15} + 8\sqrt{5} \)

\( \sqrt{6}(2\sqrt{3} - \sqrt{2}) = 2\sqrt{18} - \sqrt{12} = 6\sqrt{2} - 2\sqrt{3} \)

FOIL Method with Radicals:

Multiply two binomials containing radicals using FOIL (First, Outer, Inner, Last):

Example 1:

\( (\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5}) \)
\( = (\sqrt{3})^2 - (\sqrt{5})^2 \)
\( = 3 - 5 = -2 \)

Example 2:

\( (2 + \sqrt{3})(4 + \sqrt{3}) \)
\( = 8 + 2\sqrt{3} + 4\sqrt{3} + 3 \)
\( = 11 + 6\sqrt{3} \)

What is a Conjugate?

The conjugate of a binomial is formed by changing the sign between the terms:

Why Use Conjugates?

Multiplying by a conjugate eliminates radicals using the difference of squares formula:

\( (a + b)(a - b) = a^2 - b^2 \)

Rationalizing with Conjugates:

When the denominator has two terms with a radical, multiply by the conjugate:

Example 1:

\( \frac{6}{2 + \sqrt{3}} \)

Multiply by conjugate \( \frac{2 - \sqrt{3}}{2 - \sqrt{3}} \):

\( = \frac{6(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \)

\( = \frac{12 - 6\sqrt{3}}{4 - 3} \)

\( = \frac{12 - 6\sqrt{3}}{1} = 12 - 6\sqrt{3} \)

Example 2:

\( \frac{\sqrt{5}}{\sqrt{7} - \sqrt{2}} \)

Multiply by conjugate \( \frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} \):

\( = \frac{\sqrt{5}(\sqrt{7} + \sqrt{2})}{(\sqrt{7})^2 - (\sqrt{2})^2} \)

\( = \frac{\sqrt{35} + \sqrt{10}}{7 - 2} \)

\( = \frac{\sqrt{35} + \sqrt{10}}{5} \)

⚡ Quick Summary

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...
• Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...
• \( \sqrt[n]{a^m} = a^{m/n} \) (nth root as exponent)
• Product Property: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab} \)
• Quotient Property: \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \)
• Simplify by removing perfect power factors from radicand
• Only like radicals (same index and radicand) can be added/subtracted
• Multiply radicals: multiply coefficients and radicands separately
• Rationalize denominators: multiply by appropriate form of 1
• Use conjugates to rationalize binomial denominators with radicals
• Apply distributive property when multiplying radical expressions

📚 Key Formulas Reference