nth Roots and Rational Exponents

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Chapter 7.1

• nth Roots and Rational Exponents

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Vocabulary

• If then b is the nth root of a
• Example
• Notation
• Index (of the radical) The number n outside of
  the radical sign

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Writing nth roots as powers and powers as nth
roots

• Lets look at a familiar example a square root

Definition of square root Substitute ak for Add
exponents Set exponents equal Solve for k
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Writing nth roots as powers and powers as nth
roots

• In general, for n ? 1, any nth root can be
  written as a rational exponent in such a fashion

index
Reciprocal of index
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Real nth roots

• Let n be an integer greater than 1 and a be a
  real number
• If n is odd, a has one real nth root
• If n is even and a ? 0, then a has 2 real nth
  roots
• If n is even and a 0, then a has 1 nth root
• If n is even and a ? 0, then a has no real nth
  roots.

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Examples

• Find the indicated nth roots of a
• A. n 3, a -125 (Use since n
  is odd)
• B. n 4, a 16 (Use since n is
  even)

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What if the exponent is not of the form 1/n?

• Evalulate
• Get a little creative

Split into 2 fractions, such that 1 is of the
form 1/n
Simplify
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What if the exponent is not of the form 1/n?

• Evaluate

Recall that
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Summary

• Summary of nth roots
• 1.
• 2.
• 3.

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Example

• Solve for x

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Example

• The formula for the volume V of a regular
  dodecahedron (a solid with 12 regular pentagon
  faces) is V ? 7.66a3 where a is the length of the
  edge of the solid. Find the length of the edge
  of a dodecahedron that has a volume of 30 cubic
  feet. Round to two decimal places.