Lesson 2 of 5
Connect fractional powers to roots and keep their real domains.
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Worked example
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Evaluate and . Check: the principal square root is 5 and the cube root is 2.
Big ideas
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For a positive base, means the nth root raised to the mth power, where is reduced and n is positive. For example, .
The principal real even root is nonnegative and needs a nonnegative radicand. Odd roots accept negative radicands too.
A negative rational exponent adds a nonzero requirement. For example, needs .
, not x on all real inputs. When solving a root equation, check the original expression after raising both sides to a power.
Worked example
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Solve . Square to get , then . It is at least 2, and substitution gives the positive root .
Step-through
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Worked example: a root then a power
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At the output is ; at the denominator is zero.
Do this: Explain why the tempting shortcut fails.
Wrong idea: for every real x. At , the square root is , not −4. The all-real-input identity is .
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Flashcards
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The denominator 2 names a square root.
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What does mean for ?
The principal square root .
What does mean over the reals?
The real cube root of a.
Evaluate .
8, because the fourth root is 2 and .
Evaluate .
.
Evaluate using the real cube root.
−2.
What is the real domain of ?
[−3,infinity).
What is the real domain of ?
.
What equals for every real x?
.
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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