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Algebra 2 9-12 / Week 05 / Tuesday
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Week 05 Β· Radicals and Inverse Functions

Tuesday

Roots, powers and a fake answer
// The drip hose, forwards and backwards
⏱ about 20 min

Tuesday: Roots, Powers and a Fake Answer

Wren writes √(x + 3) = x - 3 on the chalkboard wall. "Solve it. Then tell me what you notice."

Comet squares both sides. "x + 3 = xΒ² - 6x + 9. So xΒ² - 7x + 6 = 0. That factors: x = 1 or x = 6."

"Check them," Wren says.

"x = 6: root of 9 is 3, and 6 minus 3 is 3. Good. x = 1: root of 4 is 2, but 1 minus 3 is -2."

Nova pulses softly over the 1. "Would you like a hint? A square root is never negative."

"So 2 cannot equal -2. x = 1 is extraneous," Comet says. "Squaring made it up, like clearing a denominator did last week."

"Now write √x as a power," Wren says. "x to the one half. Roots are fraction exponents in disguise."

Way one: square both sides, then check

  1. Isolate the root on one side if it is not alone already.
  2. Square both sides. The root disappears, and a polynomial equation is left.
  3. Solve the polynomial equation. It may be linear or quadratic.
  4. Check every answer in the original equation. Throw out any value where the two sides disagree.

For √(x + 3) = x - 3, squaring gives x + 3 = (x - 3)². Collect terms: x² - 7x + 6 = 0.

The solutions of that quadratic are x = 1 and x = 6. Check x = 1: √4 = 2, but 1 - 3 = -2.

A square root is never negative, so x = 1 is extraneous. Check x = 6: √9 = 3 and 6 - 3 = 3. It works.

An equation like √(x + 3) = -2 has no solution at all. Squaring gives x = 1, but the check fails, because a root cannot be negative.

Way two: think before you square

Sometimes the check comes first. In √(x + 3) = x - 3, the left side is never negative. So x - 3 must be 0 or more.

That means x must be at least 3 before you even start. The value 1 was never a candidate.

Both ways reach the same answer. Way one is mechanical and safe. Way two is quicker when you see it.

Rational exponents: roots as powers

A root is a fraction power. √x is x^(1/2), βˆ›x is x^(1/3), and βˆ›(xΒ²) is x^(2/3). The bottom is the root, the top is the power.

So 8^(2/3) means the cube root of 8, then squared: 2² = 4. And 16^(3/4) = (⁴√16)³ = 2³ = 8.

The exponent rules still hold. x^(1/2) Γ— x^(1/2) = x^(1/2 + 1/2) = x. That is exactly why √x Γ— √x = x.

Simplify roots by pulling out perfect squares: √48 = √16 Γ— √3 = 4√3, and √72 = 6√2.

Compare the two ways

EquationWay one: square and checkWay two: think firstSolution
√(x + 3) = x - 3x = 1 or 6, then the check rejects 1x - 3 must be 0 or more, so x is at least 3x = 6
√(x + 3) = -2x = 1, then the check failsa root is never negativeno solution
√(4t + 1) = 5t = 6, and the check passesthe right side is positive, so squaring is safet = 6
RADICAL EQUATIONS
  • Read the question.
  • Tap your answer.
√(x + 3) = x - 3 squares to x² - 7x + 6 = 0. Which value is a true solution of the original?
Solve √(x + 3) = -2.
Solve √(4t + 1) = 3 for t.
RATIONAL EXPONENTS
  • Read the question.
  • Tap your answer.
What is 8^(2/3)?
What is 16^(3/4)?
What is 27^(1/3)?
What is 25^(3/2)?
SIMPLIFY THE ROOTS
  • Read the question.
  • Tap your answer.
Which equals √48 in simplest form?
Which equals √72 in simplest form?
Which expression equals βˆ›(xΒ²) written with a rational exponent?
SOLVE √(X + 3) = X - 3, IN ORDER
  • ?Check x = 6: both sides equal 3, so keep it
  • ?Square both sides: x + 3 = (x - 3)Β²
  • ?Check x = 1: the root is positive but x - 3 is negative, so reject it
  • ?Expand and collect: xΒ² - 7x + 6 = 0
  • ?Factor and solve: x = 1 or x = 6
WHY THIS EXERCISEThe check is the last link of the chain. Without it, the fake answer slips through.
StatementTrue or false?
A square root of a number is never negative.?
8^(2/3) = 4?
16^(3/4) = 12?
√(x + 3) = -2 has the solution x = 1.?
4 Γ— 4 Γ— 3 = 48?
WHY THIS EXERCISERational exponents and radical equations are one topic: roots are powers, and squaring is how you undo one.
Try it
Write √(2x + 1) = x - 1 on a card. Square, solve the quadratic, then check both values.
Which value survives? Write one sentence about why the other one appeared.

Excellent. Tomorrow the stopwatch comes out in the Glass House Lab and your own drip table gets an inverse.

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