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3 of 8© Boardworks 2014 Exploring rational exponents If b 3 = a 6, write b in terms of a. b is a radical expression: b = √ a If b 3 = a, write b in terms of a. using properties of exponents, divide the exponent on both sides by 3: b 3 = a 6 but b can also be found using exponents: divide the exponent on both sides by 3: b 3 = a 1 b 3/3 = a 1/3 b 1 = a 1/3 b = a 1/3 What is the relationship between √ a and a 1/3 ?, or more generally √ a and a 1/ n ? b 3/3 = a 6/3 b 1 = a 2 b = a 2 b 3 = a √ a = a 1/ n n n 3 3
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Rational ExponentsRational Exponents
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3 of 8 © Boardworks 2014
Exploring rational exponents
If b3 = a6, write b in terms of a.
b is a radical expression: b = √a
If b3 = a, write b in terms of a.
using properties of exponents, divide the exponent on both sides by 3:
b3 = a6
but b can also be found using exponents:
divide the exponent on both sides by 3:
b3 = a1 b3/3 = a1/3 b1 = a1/3 b = a1/3
What is the relationship between √a and a1/3?, or more generally √a and a1/n?
b3/3 = a6/3 b1 = a2 b = a2
b3 = a
√a = a1/nn
n
3
3
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Combining radicals and powers
The power on the radicand becomes the numerator of the rational exponent, and the index of the radical becomes the denominator of the rational exponent.
Using what you know about exponents and radicals, write √a2 as a rational exponent. a2/3
What does it mean if the rational exponent is negative, e.g., a–2/3?
Remember that x –1 = ,x1
so a –2/3 = .a 2/3
1
A negative exponent makes the number its reciprocal.
3
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Rational exponent summary
√a = a1/n
nth root of a real number a:n
√am = a m/n
nth root of a real number a raised to the mth power:
n
n is a whole number.
n is a whole numberm is an integer, and if m < 0, a ≠ 0.
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Evaluating rational exponents
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Properties of rational exponents
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Converting between forms
