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Algebra Roots and Radicals

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Radicals (also called roots) are directly related to exponents. Roots and Radicals

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The simplest types of radicals are square roots and cube roots. Radicals beyond square roots and cube roots exist, but we will not explore them here. Roots and Radicals

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The rules for radicals that you will learn work for all radicals – not just square roots and cube roots. Roots and Radicals

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The symbol used to indicate a root is the radical symbol - Roots and Radicals

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Every radical expression has three parts… Radical symbol Index Radicand Roots and Radicals

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Every radical expression has three parts… Index Radicand Radical Roots and Radicals

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The index of a radical is a whole number greater than or equal to 2. Roots and Radicals

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The index of a square root is always 2. Roots and Radicals index

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By convention, an index of 2 is not written since it is the smallest possible index. Roots and Radicals index

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but is normally written as. The square root of 49 could be written as … Roots and Radicals

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The index of a cube root is always 3. All indices greater than 2 must be written. Roots and Radicals plural of index

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The cube root of 64 is written as Roots and Radicals.

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What does square root mean? What does cube root mean? Roots and Radicals

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The square root of a number (or expression) is another number (or expression)… Roots and Radicals …which when multiplied by itself (squared) gives back the original number (or expression). Square Root

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The cube root of a number (or expression) is another number (or expression) … Roots and Radicals …which when multiplied by itself three times (cubed) gives back the original number (or expression). Cube Root

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Roots and Radicals Example: because Also because

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Roots and Radicals Example: has two answers: 7 is called the positive or principal square root. -7 is called the negative square root.

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Roots and Radicals Example: because

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What are the first 10 whole numbers that are perfect squares? Roots and Radicals 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

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What are the first 10 whole numbers that are perfect cubes? Roots and Radicals 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

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If a number is a perfect square, then you can find its exact square root. Roots and Radicals A perfect square is simply a number (or expression) that can be written as the square [raised to 2 nd power] of another number (or expression).

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Examples: principal square root Roots and Radicals

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Examples: Roots and Radicals principal square root

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Roots and Radicals If a number is a perfect cube, then you can find its exact cube root. A perfect cube is simply a number (or expression) that can be written as the cube [raised to 3 rd power] of another number (or expression).

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Examples: Roots and Radicals principal cube root

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Examples: Roots and Radicals principal cube root

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Examples – Simplifying Square Roots: perfect square Roots and Radicals

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MultiplicationDivision b may not be equal to 0. The Rules (Properties) Roots and Radicals

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MultiplicationDivision Examples: Intermediate Algebra MTH04 Roots and Radicals

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Conjugates Radical conjugates are two expressions of the form. Conjugates have the property that when you multiply them, you get a rational number – the radical is gone. Roots and Radicals

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Intermediate Algebra MTH04 Example – Conjugates: Roots and Radicals

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Rationalizing the Denominator The process of removing a radical from the denominator of a fraction is called rationalizing the denominator. Roots and Radicals

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Rationalizing the Denominator To do this, multiply the fraction with the radical in the denominator by “1” as a fraction where the numerator and denominator are either: the radical factor that will produce a perfect square in the denominator radical or the expression that is the conjugate of the denominator of the fraction to be rationalized. Roots and Radicals

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Examples: Roots and Radicals

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