Nth Roots and Rational Exponents

Explore principal even roots, odd roots, rational exponents.

Lesson 1 of 53 parts Practice Flashcards

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Principal Even Roots

Objective: Explain principal even roots and use it to solve or interpret a problem.

The principal even root of a nonnegative real number is the nonnegative root. Even roots of negative radicands have no real values. No, it is |x| for real x.

Worked examples

Try itNth Roots and Rational Exponents — practice 1.1: Find the principal square root of 25.Show answer

Step 1: 5²=25. Step 2: The principal square root is nonnegative. Step 3: Therefore √25=5.

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Principal Even Roots

Objective: Explain principal even roots and use it to solve or interpret a problem.

The principal even root of a nonnegative real number is the nonnegative root. Even roots of negative radicands have no real values. No, it is |x| for real x.

Worked examples

Try itNth Roots and Rational Exponents — practice 1.1: Find the principal square root of 25.Show answer

Step 1: 5²=25. Step 2: The principal square root is nonnegative. Step 3: Therefore √25=5.

Odd Roots

Objective: Explain odd roots and use it to solve or interpret a problem.

Odd roots accept negative real radicands and preserve their sign. A cube-root function has all real inputs and outputs. Because (−2)³=−8.

Worked examples

Try itNth Roots and Rational Exponents — practice 2.1: Find the real cube root of −512.Show answer

Step 1: (−8)³=−512. Step 2: Odd roots preserve the negative sign. Step 3: The answer is −8.

Rational Exponents

Objective: Explain rational exponents and use it to solve or interpret a problem.

For suitable real domains, a^(m/n) means an nth root followed by an mth power. Negative bases require careful attention to the root index and notation. An even root of a negative base is not real.

Worked examples

Try itNth Roots and Rational Exponents — practice 3.1: Evaluate 121^(1/2) using the principal root.Show answer

Step 1: Exponent 1/2 means principal square root. Step 2: √121=11. Step 3: The output is 11.

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