Algebra 2: Square Roots and Radicals

• Square roots and radicals are fundamental concepts in algebra, particularly in Algebra 2, with nth roots generalizing the idea of square roots.

Square Roots

• The square root of a number ( x ) is a value that, when multiplied by itself, equals ( x ).
• ( \sqrt{x} ) denotes the principal (non-negative) square root of ( x ).
• Example: ( \sqrt{9} = 3 ) because ( 3 \times 3 = 9 ).
• For real numbers, the square root of a negative number is not a real number (it is an imaginary number).
• The square root function ( f(x) = \sqrt{x} ) is defined for ( x \geq 0 ) in the real number system.

Radicals

• A radical is an expression that includes a root, such as a square root, cube root, etc.
• The general form of a radical is ( \sqrt[n]{x} ), where ( n ) is the index (the degree of the root) and ( x ) is the radicand (the value under the radical).
• When ( n = 2 ), the radical represents a square root, and the index is often omitted: ( \sqrt{x} \equiv \sqrt[2]{x} \equiv \sqrt{x} ).
• ( \sqrt[3]{x} ) represents the cube root of ( x ).

Properties of nth Roots

• The nth root of a number ( x ) is a value that, when raised to the nth power, equals ( x ).
• ( \sqrt[n]{x} ) represents the nth root of ( x ).
• Example: ( \sqrt[3]{8} = 2 ) because ( 2^3 = 8 ).
• If ( n ) is even, ( x ) must be non-negative for real number results.
• If ( n ) is odd, ( x ) can be any real number.
• The nth root can also be expressed as a rational exponent: ( \sqrt[n]{x} = x^{\frac{1}{n}} ).

Properties of Radicals

• Product Property: ( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} ) for non-negative ( a ) and ( b ) (when ( n ) is even).
• Quotient Property: ( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} ) for ( b \neq 0 ) and non-negative ( a ) and ( b ) (when ( n ) is even).
• Power of a Radical: ( (\sqrt[n]{a})^m = \sqrt[n]{a^m} = a^{\frac{m}{n}} ).
• Root of a Root: ( \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} ).

Simplifying Radicals

• Simplifying radicals involves removing all perfect nth powers from the radicand.
• Example: ( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} ).
• To simplify ( \sqrt[3]{24} ), find the largest perfect cube that divides 24, which is 8; thus, ( \sqrt[3]{24} = \sqrt[3]{8 \cdot 3} = \sqrt[3]{8} \cdot \sqrt[3]{3} = 2\sqrt[3]{3} ) simplifies to ( \sqrt{24} = \sqrt{8 \cdot 3} = \sqrt{8} \cdot \sqrt{3} = 2\sqrt{3} ).
• Rationalizing the denominator involves eliminating radicals from the denominator of a fraction.
• Multiply the numerator and denominator by a radical that will eliminate the radical in the denominator.
• Example: To rationalize ( \frac{1}{\sqrt{2}} ), multiply by ( \frac{\sqrt{2}}{\sqrt{2}} ) to get ( \frac{\sqrt{2}}{2} ).

Inverse Properties of nth Powers and nth Roots

• The nth power and nth root are inverse operations of each other.
• Inverse property: ( (\sqrt[n]{x})^n = x ) and ( \sqrt[n]{x^n} = x ) if ( n ) is odd, or ( |x| ) if ( n ) is even.
• Example: ( (\sqrt[3]{5})^3 = 5 ) and ( \sqrt[3]{5^3} = 5 ).
• Example for even roots: ( \sqrt{x^2} = |x| ); for instance, ( \sqrt{(-3)^2} = \sqrt{9} = 3 = |-3| ).
• The composition of a function and its inverse results in the original input.
• If ( f(x) = x^n ) and ( g(x) = \sqrt[n]{x} ), then ( f(g(x)) = (\sqrt[n]{x})^n = x ) and ( g(f(x)) = \sqrt[n]{x^n} = x ) (for odd ( n ) or non-negative ( x ) if ( n ) is even).

Rational Exponents

• A rational exponent is an exponent that is a rational number (can be expressed as a fraction).
• ( x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m ).
• Example: ( 8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 ) or ( 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = (2)^2 = 4 ) is shown as ( 8^{\frac{2}{3}} = \sqrt{8^2} = \sqrt{64} = 4 ) or ( 8^{\frac{2}{3}} = (\sqrt{8})^2 = (2)^2 = 4 ).
• Negative rational exponents: ( x^{-\frac{m}{n}} = \frac{1}{x^{\frac{m}{n}}} ).
• Example: ( 4^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}} = \frac{1}{\sqrt{4}} = \frac{1}{2} ).

Complex Numbers and Radicals

• When dealing with square roots of negative numbers, complex numbers are introduced.
• ( \sqrt{-1} ) is defined as the imaginary unit, denoted by ( i ).
• ( i^2 = -1 ).
• Example: ( \sqrt{-9} = \sqrt{9 \cdot -1} = \sqrt{9} \cdot \sqrt{-1} = 3i ).
• Complex numbers have the form ( a + bi ), where ( a ) and ( b ) are real numbers.
• Operations with complex numbers:
  • Addition/Subtraction: Combine like terms; ( (a + bi) + (c + di) = (a + c) + (b + d)i ).
  • Multiplication: Use the distributive property and remember ( i^2 = -1 ); ( (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i ).

Solving Radical Equations

• A radical equation is an equation that contains a radical expression with a variable in the radicand.
• To solve radical equations:
  • Isolate the radical on one side of the equation.
  • Raise both sides of the equation to the power equal to the index of the radical.
  • Solve for the variable.
  • Check for extraneous solutions (solutions that satisfy the transformed equation but not the original).
• Example: Solve ( \sqrt{x - 2} = 3 ). Square both sides: ( (\sqrt{x - 2})^2 = 3^2 ), so ( x - 2 = 9 ). Then ( x = 11 ). Check: ( \sqrt{11 - 2} = \sqrt{9} = 3 ), so ( x = 11 ) is a valid solution.
• Example with extraneous solutions: Solve ( \sqrt{x + 3} = x - 3 ). Square both sides: ( x + 3 = (x - 3)^2 = x^2 - 6x + 9 ). Rearrange to ( x^2 - 7x + 6 = 0 ). Factor: ( (x - 6)(x - 1) = 0 ), so ( x = 6 ) or ( x = 1 ). Check ( x = 6 ): ( \sqrt{6 + 3} = \sqrt{9} = 3 ) and ( 6 - 3 = 3 ), so ( x = 6 ) is a solution. Check ( x = 1 ): ( \sqrt{1 + 3} = \sqrt{4} = 2 ) and ( 1 - 3 = -2 ), so ( x = 1 ) is an extraneous solution.