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Algebra 1 9-12 / Week 07 / Tuesday
2/6
Week 07 · Exponents and Exponential Change

Tuesday

Roots as exponents
// Linear change adds, exponential change multiplies
⏱ about 20 min

Tuesday: Roots as Exponents

Wren writes 2³ × 2⁴ on the whiteboard. "Three 2s times four 2s. What do you notice?"

"Seven 2s," Comet says. "2⁷. Add the exponents. We did this in Expressions and Equations."

"And (2³)⁴?" he asks. "Four groups of three 2s. 2¹². Multiply the exponents."

Nova lowers a card into the light. It says 8 to the power one third. "Would you like a hint? Use the second rule on it."

"Cube it," Comet says slowly. "One third times 3 is 1. So the cube of that thing is 8 to the 1. Just 8."

"Then 8 to the one third is the number whose cube is 8," Wren says. "The cube root. Two."

"A root is a power with a fraction on top," Comet says. "Our engineer, does that work for square roots too?"

Two rules you already know

  • Product rule: am × an = am + n. Same base, add the exponents.
  • Power rule: (am)n = amn. A power of a power multiplies the exponents.
  • Both rules just count factors. 2³ × 2⁴ has 3 + 4 = 7 factors of 2.

What a fraction exponent must mean

Nobody can multiply 8 by itself one third of a time. So the crew asks the rules to decide.

Suppose the power rule still holds. Then (8 to the 1/3) cubed is 8 to the (1/3) × 3. That exponent is 1, so the cube is 8.

So 8 to the 1/3 is the number whose cube is 8. That is the cube root of 8, and it equals 2.

The same reasoning gives 9 to the 1/2 as the square root of 9. The bottom of the fraction is a root.

A top number is a power. 16 to the 3/4 is the fourth root of 16, cubed: 2 cubed, which is 8.

PowerRead asValue
9^(1/2)the square root of 93
8^(1/3)the cube root of 82
27^(2/3)the cube root of 27, then squared9
16^(3/4)the fourth root of 16, then cubed8
32^(1/5)the fifth root of 322

Two ways to the same answer

For 27 to the 2/3, Comet takes the root first: the cube root of 27 is 3, and 3 squared is 9.

Wren squares first: 27 squared is 729, and the cube root of 729 is 9.

Both ways agree. Root first keeps the numbers small, so the crew prefers it.

Radicals rewrite the same way. The square root of x is x to the 1/2. The cube root of x squared is x to the 2/3.

RULES OF EXPONENTS
  • Read the question.
  • Tap your answer.
2³ × 2⁴ is 2 to what power?
5⁶ ÷ 5² is 5 to what power?
What is 3⁴?
(2³)⁴ is 2 to what power?
WHY 8 TO THE 1/3 IS THE CUBE ROOT OF 8
  • ?The number whose cube is 8 is the cube root of 8, which is 2.
  • ?(1/3) × 3 = 1, so the cube equals 8 to the 1, which is 8.
  • ?Cube it: (8 to the 1/3) cubed = 8 to the (1/3) × 3.
  • ?Assume the power rule still works for the fraction 1/3.
WHY THIS EXERCISEThe meaning of a fraction exponent is not a new fact to memorize. It follows from keeping the old rules true.
RATIONAL EXPONENTS AND RADICALS
  • Read the question.
  • Tap your answer.
What is 8^(1/3)?
What is 27^(2/3)?
Which power equals the square root of x?
Which equals √50 in simplest form?
StatementTrue or false?
9^(1/2) = 3?
8^(1/3) = 4?
16^(3/4) = 8?
2⁷ = 128?
The square root of x can be written as x to the power 1/2.?
WHY THIS EXERCISERational exponents let roots follow the same rules as whole-number powers, so one set of rules covers everything.
Try it
Write 64 to the 1/2, 64 to the 1/3 and 64 to the 1/6 on three cards. Work out each one.
Check each by raising your answer back to the power 2, 3 or 6. It should return to 64.
Draw a number line from 0 to 10. Mark 64 to the 1/2, 64 to the 1/3 and 64 to the 1/6.

Excellent reasoning. Tomorrow a rubber ball bounces in the Loft Lab, and its heights shrink by the same factor every time.

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