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Prealgebra / LEVEL 10 · DIFFICULTY 4/5

Square Roots & Exact Values

Estimate roots, simplify radical structure and keep exact quantities.

3 stages · 24 practice problems · two 6-question assessment forms

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  1. MINI QUEST 10.1Square Roots & BoundsRead the lesson
  2. MINI QUEST 10.2Extracting Square FactorsRead the lesson
  3. MINI QUEST 10.3Combining & Multiplying RadicalsRead the lesson
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STAGE 10.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Square Roots & Bounds

Useful preparation: Powers, Signs & Grouping

Goal: Interpret the principal square root and bound it with perfect squares.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A square root reverses squaring with a sign convention

The symbol √a means the nonnegative square root for a ≥ 0. An equation x² = a can have two real solutions even though √a names just one.

√49 = 7; x² = 49 gives x = ±7

Neighboring squares give bounds

Squaring preserves order among nonnegative numbers. Place a number between consecutive squares to locate its root between consecutive integers.

64 < 70 < 81 ⇒ 8 < √70 < 9

Exact and approximate values serve different purposes

A radical can be an exact answer. A decimal approximation should state its precision; do not replace the exact value prematurely.

√2 ≈ 1.414
WORKED EXAMPLE 1

Between which consecutive integers is √115?

  1. 10² = 100 and 11² = 121.
  2. 115 lies strictly between those squares.

10 and 11

WORKED EXAMPLE 2

Evaluate √((-9)²).

  1. The square is 81.
  2. The principal square root of 81 is nonnegative.

9

Common pitfalls

Possible mix-up: √36 is both 6 and −6.

The radical denotes 6; the equation x² = 36 has both solutions.

Possible mix-up: √(a²) always equals a.

For real a it equals |a|.

Explain it to yourself

Explain how a square picture and a number-line estimate describe the same root.

Preview the eight practice prompts
  1. Compute √169.
  2. Compute √((-12)²).
  3. What is the greatest integer less than √55?
  4. What is the least integer greater than √130?
  5. How many real solutions does x² = 81 have?
  6. Is √50 greater than 7.1? Enter yes or no.
  7. A square garden has area 90 m². What is the smallest whole-meter length that is at least as long as one side? New context
  8. A square screen has area 200 square units. Is its side longer than 14 units? Enter yes or no. New context
Open stage 10.1 in the student workspace →

STAGE 10.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Extracting Square Factors

Useful preparation: Square Roots & Bounds · Prime Factorization

Goal: Separate a perfect-square factor from the remaining radical.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Use products of nonnegative factors

For nonnegative a and b, √(ab) = √a √b. A perfect-square factor can therefore move outside the radical as its root.

√72 = √(36 × 2) = 6√2

Pair prime factors

Each pair of identical prime factors makes a square. Take one factor outside for each pair and leave any unpaired factors inside.

√(2³ × 3²) = 2 × 3√2

Stop when the inside has no square factor

An equivalent radical is fully simplified when its positive integer radicand has no square factor above one. Different first factorizations should reach the same result.

2√18 = 6√2
WORKED EXAMPLE 1

Simplify √147.

  1. Factor 147 as 49 × 3.
  2. The square root of 49 is 7.

7√3

WORKED EXAMPLE 2

Simplify √(25/49).

  1. The numerator and denominator are nonnegative squares.
  2. Take their nonnegative roots separately.

5/7

Common pitfalls

Possible mix-up: √(a + b) equals √a + √b.

The product rule does not distribute over addition.

Possible mix-up: √48 = 4√3 cannot be checked.

Square 4√3: 16 × 3 = 48.

Explain it to yourself

Why do two different square-factor choices lead to the same simplified radical?

Preview the eight practice prompts
  1. Write √75 = a√3 with a > 0. Find a.
  2. Write √98 = a√2 with a > 0. Find a.
  3. Write √180 = 6√b where b is a positive integer. Find b.
  4. Compute √(16/81).
  5. Write 2√45 = a√5 with a > 0. Find a.
  6. Write √288 = a√2 with a > 0. Find a.
  7. A square has area 108 square units. Its side is a√3 units with a > 0. Find a. New context
  8. A square tile has area 242 cm². Its side is a√2 cm with a > 0. What is the integer a? New context
Open stage 10.2 in the student workspace →

STAGE 10.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Combining & Multiplying Radicals

Useful preparation: Extracting Square Factors · Expressions & Substitution

Goal: Recognize like radicals and preserve exact values.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Like radicals combine like like terms

Terms with the same simplified radicand share a common factor. Add their coefficients while keeping the radical factor.

3√2 + 5√2 = 8√2

Simplify before deciding whether terms match

Radicals that look different may have the same square-free radicand after extracting squares.

√12 + √27 = 2√3 + 3√3

Products can remove a radical

Multiply coefficients and nonnegative radicands separately. In particular, a principal square root times itself gives its radicand.

(a√b)(c√d) = ac√(bd)
WORKED EXAMPLE 1

Simplify √20 + 2√45.

  1. √20 = 2√5 and 2√45 = 6√5.
  2. Combine the coefficients 2 and 6.

8√5

WORKED EXAMPLE 2

Compute (3√2)(4√8).

  1. Multiply to get 12√16.
  2. √16 = 4.

48

Common pitfalls

Possible mix-up: √2 + √3 equals √5.

The product rule does not work for sums.

Possible mix-up: 3√5 + 2√5 equals 5√10.

Keep the shared factor √5: the sum is 5√5.

Explain it to yourself

Which operation can make two irrational factors produce a rational result?

Preview the eight practice prompts
  1. Write 4√3 + 7√3 = a√3. Find a.
  2. Write √8 + √18 = a√2. Find a.
  3. Compute √6 × √24.
  4. Compute (2√5)(3√5).
  5. Write 3√12 − √27 = a√3. Find a.
  6. Compute √75 ÷ √3.
  7. A rectangle has sides 2√3 and 5√3 meters. What is its area in square meters? New context
  8. A path has segments √50, √8 and √18 meters. Its total length is a√2 meters. Find a. New context
Open stage 10.3 in the student workspace →
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