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Lesson · Grade 11 · Pre Calculus

Rational exponents: a fraction power is a root

  • Grade 11
  • Pre Calculus
  • 1 min 59 s
  • British Columbia
  • BC11-MA-65

Three copies of 8^(1/3) multiply to 8^1 = 8, so 8^(1/3) is the cube root of 8; then 8^(2/3) = (∛8)² = 2² = 4, which shows that in a rational exponent the bottom of the fraction is the root and the top is the power. The answer is checked by squaring first (∛64 = 4), taking the root first is shown to keep the numbers small, and the common mistake of reading 8^(2/3) as 8 × 2/3 is ruled out.

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In one sentence

The question this lesson answers

What does 8 to the power of two thirds mean?

8 is the base, and the small raised number, two thirds, is the exponent.

Transcript

What the lesson says

What does 8 to the power of two thirds mean?

8 is the base, and the small raised number, two thirds, is the exponent.

An exponent that is a fraction is called a rational exponent.

Start simpler: multiply 8 to the one third by itself three times.

Multiplying powers of the same base adds the exponents: one third plus one third plus one third is 1, and 8 to the power of 1 is 8.

So 8 to the one third is the number whose three copies multiply to 8: the cube root of 8.

The cube root of 8 is 2, because 2 times 2 times 2 is 8.

The same holds for any positive x: x to the one half is its square root, and x to the one third is its cube root.

So the bottom of the fraction, the denominator, tells you which root to take.

Back to two thirds: that is one third times 2, and a power of a power multiplies the exponents.

So 8 to the two thirds is the cube root of 8, squared. The top of the fraction is the power.

The cube root of 8 is 2, and 2 squared is 4.

Check by squaring first: 8 squared is 64, and the cube root of 64 is 4, because 4 times 4 times 4 is 64.

Both ways give 4, but taking the root first keeps the numbers small.

A common mistake is to read it as 8 times two thirds. That gives sixteen thirds, which is not 4.

In a rational exponent, the bottom is the root and the top is the power: 8 to the two thirds is 4.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Real numbers and powers: powers with rational exponentsBC11-MA-65

Previous: Simplifying with exponent laws and negative exponentsNext: Simplifying, Adding and Multiplying Radicals

The outcomes and strands follow the British Columbia curriculum.

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