Unit 2 · NumbersN3 · Exponents & Radicals · Propulsion

Radicals & Rational Exponents

Simplify radicals, translate rational exponents, and preserve roots and domains.

29 min
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Desmos way

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7 steps · about 29 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

Radicand: The expression inside a radical symbol.
Principal root: The nonnegative square root.
Rational exponent: An exponent whose denominator names a root.

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

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The big idea

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See the structure

Factor the radicand into a perfect-square part and what remains. Thus $\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2$ .

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

The exponent $m/n$ means take the $n$ th root and raise to the $m$ th power. For real even roots, the radicand must be nonnegative.

Explore in Desmos

Separate a principal root from equation roots

The expression

\sqrt{16}

returns the principal value 4. The graph intersection shows that the equation

x^2=16

has two solutions,

-4

and 4. Explain why these are different questions.

The calculator is live: change anything and see what happens.

Separate a principal root from equation roots

Degree mode

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Stretch to a 36

You may combine like radicals only. $2\sqrt3+5\sqrt3=7\sqrt3$ , while $\sqrt2+\sqrt3$ does not simplify.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Extract the largest perfect-square factor

Problem

Which expression is equivalent to

\sqrt{72}

in simplest radical form?

Factor

72=36\cdot2

Which radical step preserves the exact value?

\sqrt{72}=\sqrt{36}\sqrt2=6\sqrt2

Which square or sign check confirms the radical result?

Worked example

Example 2: Use principal-root meaning

Problem

What is the value of

\sqrt{(-11)^2}

The square is 121, and the principal square root is nonnegative.

Which radical step preserves the exact value?

\sqrt{121}=11

, equivalently

\sqrt{x^2}=|x|

Which square or sign check confirms the radical result?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

Which expression is equivalent to

\sqrt{180}

in simplest radical form?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Numerically verify an exact radical form

Type sqrt(180) and 6sqrt(5). Their matching decimal displays show that the simplified radical has the same value.

Type (6sqrt(5))^2 and read 180. Desmos verifies equivalence; algebraic factoring

180=36\cdot5

produces the exact form

6\sqrt5

Numerically verify an exact radical form

Degree mode

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.