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Roots, Radicals, and Complex Numbers

Chapter 7. Roots, Radicals, and Complex Numbers. Chapter Sections. 7.1 – Roots and Radicals 7.2 – Rational Exponents 7.3 – Simplifying Radicals 7.4 – Adding, Subtracting, and Multiplying Radicals 7.5 – Dividing Radicals 7.6 – Solving Radical Equations 7.7 – C omplex Numbers.

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Roots, Radicals, and Complex Numbers

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  1. Chapter 7 Roots, Radicals, and Complex Numbers

  2. Chapter Sections 7.1 – Roots and Radicals 7.2 – Rational Exponents 7.3 – Simplifying Radicals 7.4 – Adding, Subtracting, and Multiplying Radicals 7.5 – Dividing Radicals 7.6 – Solving Radical Equations 7.7 – Complex Numbers

  3. Simplifying Radicals § 7.3

  4. Understand Perfect Powers Perfect Power, Perfect Square, Perfect Cube • A perfect power is a number or expression that can be written as an expression raised to a power that is a whole number greater than 1. • A perfect square is a number or expression that can be written as a square of an expression. A perfect square is a perfect second power. • A perfect cube is a number or expression that can be written as a cube of an expression. A perfect cube is a perfect third power.

  5. Perfect Powers This idea can be expanded to perfect powers of a variable for any radicand. A quick way to determine if a radicand xmis a perfect power for an index is to determine if the exponent mis divisible by the index of the radical. Since the exponent, 20, is divisible by the index, 5, x20 is a perfect fifth power. Example:

  6. Product Rule for Radicals Product Rule for Radicals For nonnegative real numbers a and b, Example: √20 can be factored into any of these forms.

  7. Product Rule for Radicals To Simplify Radicals Using the Product Rule • If the radicand contains a coefficient other than 1, if possible, write it as a product of two numbers, one of which is the largest perfect power for the index. • Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index. • Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers (numbers and variables) under the same radical. • Simplify the radical containing the perfect powers.

  8. Product Rule for Radicals Examples:

  9. Quotient Rule for Radicals Quotient Rule for Radicals For nonnegative real numbers a and b, Examples:

  10. Quotient Rule for Radicals Examples:

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