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Chapter 5: Exponential and Logarithmic Functions 5.1: Radicals and Rational Exponents Essential Question: Explain the meaning of using radical expressions

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5.1: Radicals and Rational Exponents BACKGROUND INFO (NO NEED TO COPY) Recall that when x 2 = c (some constant), there were two solutions, and, when the constant was positive. You had no solutions when the constant was negative. x 2 = 9 x = 3 or x = -3 When x 3 = c, there was only one solution,, and the answer was positive or negative depending on if x was positive or negative to start. x 3 = 64 x = 4 x 3 = -64 x = -4 All other higher roots act in a similar fashion

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5.1: Radicals and Rational Exponents Solutions to x n = c When n (exponent) is even If c > 0, one positive and one negative solution If c = 0, one solution (x = 0) If c < 0, no solution When n is odd One solution Recall help: Even # power = even # of roots Odd # power = odd # of roots

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5.1: Radicals and Rational Exponents n th roots The n th root of c is denoted by either of the symbols: Rules about n th roots If the outside root is the same, numbers underneath can be multiplied and divided If the number underneath the root is the same, they are like terms, and can be added or subtracted

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5.1: Radicals and Rational Exponents Example 1: Operations on n th roots Example 2: Evaluating n th roots Use a calculator to approximate each expression the nearest ten thousandth., entered as 40^(1/5) , entered as 225^(1/11)

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5.1: Radicals and Rational Exponents Rational Exponents Rational exponents of the form are called n th roots. Rational exponents can also be written in the form. The numerator of the exponent represents the power a base is taken to. The denominator of the exponent represents the root. The order of application does not matter

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5.1: Radicals and Rational Exponents Assignment Page 334 Problems 1 – 37, odd problems Ignore the instructions about not using a calculator in problems 1 – 15 Make sure to simplify your square roots Show all non-calculator work Due tomorrow

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Chapter 5: Exponential and Logarithmic Functions 5.1: Radicals and Rational Exponents (Day 2) Essential Question: Explain the meaning of using radical expressions

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5.1: Radicals and Rational Exponents Laws of Exponents (A recap) c r c s = c r+s Multiplying same base == add exponents = c r-s Dividing same base == subtract exponents (c r ) s = c rs Power to power == multiply exponents (cd) r = c r d r and Power on outside == multiply exponents to all bases c -r = Negative exponents move to other side of the fraction and become positive

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5.1: Radicals and Rational Exponents Simplifying Expressions with Rational Exponents (Ex 3) Write the expression using only positive exponents Distribute the 2/3 on the outside Simplify the coefficient part & move negative exponents

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5.1: Radicals and Rational Exponents Simplifying Expressions with Rational Exponents Example 4: Distribute… and since bases are shared, add the exponents Example 5: Take the -2 power first, then add exponents to like bases (as in Ex 4 above)

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5.1: Radicals and Rational Exponents Simplifying Expressions with Rational Exponents Ex 6: Write the expression without using radicals, using only positive exponents Get rid of the radicals. Root = denominator Multiply powers of powers. Add exponents of common bases

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5.1: Radicals and Rational Exponents Rationalizing the Numerator/Denominator Rationalizing means removing all roots from the specified side of a fraction Simple roots can be removed by multiplying top/bottom by the root. Complex roots can be removed by multiplying with the conjugate Rationalizing a numerator works the same as above

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5.1: Radicals and Rational Exponents Assignment Page 334 Problems 39-77, odd problems Make sure to simplify your square roots Show all non-calculator work Due whenever we get back

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