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• Emman Sisnorio How do I solve if it is only exponents and variables? 11 months ago
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• Definitions A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares. A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.
• Perfect Powers A quick way to determine if a radicand x n is a perfect power for an index is to determine if the exponent n is divisible by the index of the radical. Example : Since the exponent, 20, is divisible by the index, 5, x 20 is a perfect fifth power. This idea can be expanded to perfect powers of a variable for any radicand. The radicand x n is a perfect power when n is a multiple of the index of the radicand.
• Product Rule for Radicals Examples:
• If the radicand contains a coefficient other than 1, write it as a product of the 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 under the same radical.
• Simplify the radical containing the perfect powers.
To Simplify Radicals Using the Product Rule
• Product Rule for Radicals Examples: *When the radical is simplified, the radicand does not have a variable with an exponent greater than or equal to the index.
• Quotient Rule for Radicals More Examples:
• Combine like radicals (if there are any).
• CAUTION! The product rule does not apply to addition or subtraction! 
• Multiplying Radicals Multiply: Use the FOIL method. Notice that the inner and outer terms cancel.
• Rationalizing Denominators Examples : To Rationalize a Denominator Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power. Cannot be simplified further.
• Conjugates When the denominator of a rational expression is a binomial that contains a radical, the denominator is rationalized. This is done by using the conjugate of the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed. The conjugate of The conjugate of
• Simplifying Radicals Simplify by rationalizing the denominator:
• Simplifying Radicals A Radical Expression is Simplified When the Following Are All True
• No perfect powers are factors of the radicand and all exponents in the radicand are less than the index.
• No radicand contains a fraction.
• No denominator contains a radical.
• Rational Exponents
• Changing a Radical Expression A radical expression can be written using exponents by using the following procedure: When a is nonnegative, n can be any index. When a is negative, n must be odd.
• Changing a Radical Expression Exponential expressions can be converted to radical expressions by reversing the procedure. When a is nonnegative, n can be any index. When a is negative, n must be odd.
• Simplifying Radical Expressions This rule can be expanded so that radicals of the form can be written as exponential expressions. For any nonnegative number a , and integers m and n, Power Index
• Rules of Exponents The rules of exponents from Section 5.1 also apply when the exponents are rational numbers . For all real numbers a and b and all rational numbers m and n , Product rule: a m • a n = a m + n Quotient rule: Negative exponent rule:
• Rules of Exponents For all real numbers a and b and all rational numbers m and n , Zero exponent rule: a 0 = 1, a  0 Raising a power to a power: Raising a product to a power : Raising a quotient to a power :
• Rules of Exponents Examples : 1.) Simplify x -1/2 x -2/5 . 2.) Simplify ( y -4/5 ) 1/3 . 3.) Multiply –3 a -4/9 (2 a 1/9 – a 2 ).
• Factoring Expressions Examples : 1.) Factor x 1/4 – x 5/4 . x 1/4 – x 5/4 = x 1/4 (1 – x 5/4-(1/4) ) The smallest of the two exponents is 1/4 . Original exponent Exponent factored out = x 1/4 (1 – x 4/4 ) = 2.) Factor x -1/2 + x 1/2 . x -1/2 + x 1/2 = x -1/2 (1– x 1/2 - (-1/2) ) = x -1/2 (1– x ) = Original exponent Exponent factored out The smallest of the two exponents is - 1/2 . x 1/4 (1 – x )