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MTH 092 Section 15.1 Introduction to Radicals Section 15.2 Simplifying Radicals

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Square Roots The square root of a number is another number such that, if you square the second number (multiply that number times itself) you get the first number. For example, 5 is the square root of 25 because 5 2 = 5 x 5 = 25. Most scientific calculators are equipped with a square root key. You cannot take a square root of a negative number.

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Cube Roots The cube root of a number is another number such that, if you cube the second number (multiply that number times itself three times) you get the first number. For example, 6 is the cube root of 216 because 6 3 = 6 x 6 x 6 = 216. Some scientific calculators have a cube root key.

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Fourth Roots The fourth root of a number is another number such that, if you raise the second number to the fourth power (multiply that number times itself four times) you get the first number. For example, 7 is the fourth root of 2401 because 7 4 = 7 x 7 x 7 x 7 = 2401. I have never seen a scientific calculator with a fourth root key. You cannot take a fourth root of a negative number.

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Root Notation

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Parts of a Radical Expression

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Examples

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Roots and Variables To find a square root of a variable, divide the exponent by 2. To find a cube root of a variable, divide the exponent by 3 To find a fourth root of a variable, divide the exponent by 4

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Examples

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All Together Now

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Simplifying Square Roots First off, let’s review the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,… x 2, x 4, x 6, x 8, x 10, x 12, x 14,… The object is to find the biggest number on that list that will divide the number underneath the radical. The square root of that number will come out. The other number will stay underneath.

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Continued To find the square root of a variable, divide the exponent by 2. The quotient comes out. If the exponent is odd, there will be one left over. Leave it under the radical.

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Examples

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Simplifying Cube Roots Remember the perfect squares? Here are the perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,… The object is to find the biggest number on that list that will divide the number underneath the radical. The cube root of that number will come out. The other number will stay underneath.

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Continued To find the cube root of a variable, divide the exponent by 3. The quotient comes out. The remainder stays under the radical.

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Examples

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Exponents and Powers Power – the result of raising a base to an exponent. Ex. 32 Base – the number being raised to the exponent. Ex. 32 3 is the base.

Exponents and Powers Power – the result of raising a base to an exponent. Ex. 32 Base – the number being raised to the exponent. Ex. 32 3 is the base.

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