roots and fractional exponents. you know a square root means a number you take times itself to get a...
TRANSCRIPT
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Roots and Fractional Exponents
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You know a square root meansa number you take times itself to get a given answer.
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Square roots were named because they were the lengths of the sides of squares with a given area.
For instancea square with area 16has a sidewith length 4.
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You also know that every positive number has 2 real number square roots.
While normally we just care about the positive square root, remember there is also an equivalent negative answer.
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There are lots of other kinds of roots besides square roots.
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A cube root is the number you take times itself and then times itself again to get a given answer.
It is also the side of a cube with a given volume.
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Each number has just 1 real number cube root.
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The cube root of a positive
number is positive. The cube root of a
negative number is negative.
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Fourth roots, fifth roots, sixth roots, etc. all work similarly to square roots and cube roots.
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You can take even roots of any positive number. You will get 2 real number answers.
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You can take odd roots of any real number. You will get 1 real number answer.
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Simplifying Root Expressions with Variables Take the root of the
coefficient. Divide the exponent by the
index of the root.
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If you don’t get an even answer when you divide … Put the quotient outside
the root symbol. Put the remainder as the
exponent under the root symbol.
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Simplify
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Simplify
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Simplify
Remember square root meansthe 2nd root.
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Simplify
Technically the best answer is
Unless it specifies that multiple answers are needed, though, we usually just give the positive answer.
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For now we won’t worry about simplifying roots that don’t come out even.
However, you can estimate where an answer is by placing it between numbers whose roots you know.
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For instance …
13 is between 9 and 16 (and slightly closer to 16 than 9)
So must be between 3 and 4 (and slightly closer to 4 than 3) … Estimate 3.6
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For instance …
30 is just a little more than 27 (and much less than 64)
So must be a little more than 3 (and much less than 4)
… Estimate 3.1
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Fractional Exponents
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A fractional power with a numerator or 1 is the same as the corresponding root.
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So the original problem …
… the answer is 5.
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What is ?
Find all answers.
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What is ?
Find all answers.
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What is ?
Find all answers.
31
64
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What is ?
Find all answers.
(There is only 1 answer.)
31
64
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In a fractional exponent, the numerator (top) is a
power the denominator (bottom)
is a root
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Write as a root, and then simplify:
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Write as a root, and then simplify:
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Write using exponents.