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← Prealgebra levels 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
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 \displaystyle \sqrt{a} a means the nonnegative square root for a ≥ 0 \displaystyle a \ge 0 a ≥ 0 . An equation x 2 = a \displaystyle x^{2} = a x 2 = a can have two real solutions even though a \displaystyle \sqrt{a} a names just one.
49 = 7 \displaystyle \sqrt{49} = 7 49 = 7 ; x 2 = 49 \displaystyle x^{2} = 49 x 2 = 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 \displaystyle 64 < 70 < 81 \Rightarrow 8 < \sqrt{70} < 9 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 \displaystyle \sqrt{2} \approx 1.414 2 ≈ 1.414 WORKED EXAMPLE 1 Between which consecutive integers is 115 \displaystyle \sqrt{115} 115 ? 10 2 = 100 \displaystyle 10^{2} = 100 1 0 2 = 100 and 11 2 = 121 \displaystyle 11^{2} = 121 1 1 2 = 121 . 115 lies strictly between those squares. 10 and 11
WORKED EXAMPLE 2 Evaluate ( − 9 ) 2 \displaystyle \sqrt{\left(-9\right)^{2}} ( − 9 ) 2 . The square is 81. The principal square root of 81 is nonnegative. 9
Common pitfalls Possible mix-up: 36 \displaystyle \sqrt{36} 36 is both 6 and −6.
The radical denotes 6; the equation x 2 = 36 \displaystyle x^{2} = 36 x 2 = 36 has both solutions.
Possible mix-up: a 2 \displaystyle \sqrt{a^{2}} a 2 always equals a.
For real a it equals ∣ a ∣ \displaystyle \left|a\right| ∣ a ∣ .
Explain it to yourself Explain how a square picture and a number-line estimate describe the same root.
Preview the eight practice prompts Problem 1
Compute 169 \displaystyle \sqrt{169} 169 .
Problem 2
Compute ( − 12 ) 2 \displaystyle \sqrt{\left(-12\right)^{2}} ( − 12 ) 2 .
Problem 3
What is the greatest integer less than 55 \displaystyle \sqrt{55} 55 ?
Problem 4
What is the least integer greater than 130 \displaystyle \sqrt{130} 130 ?
Problem 5
How many real solutions does x 2 = 81 \displaystyle x^{2} = 81 x 2 = 81 have?
Problem 6
Is 50 \displaystyle \sqrt{50} 50 greater than 7.1? Enter yes or no.
Problem 7New context
A square garden has area 90 m 2 \displaystyle 90\, m^{2} 90 m 2 . What is the smallest whole-meter length that is at least as long as one side?
Problem 8New context
A square screen has area 200 square units. Is its side longer than 14 units? Enter yes or no.
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, a b = a b \displaystyle \sqrt{ab} = \sqrt{a}\, \sqrt{b} ab = a b . A perfect-square factor can therefore move outside the radical as its root.
72 = 36 × 2 = 6 2 \displaystyle \sqrt{72} = \sqrt{36 \times 2} = 6\, \sqrt{2} 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.
2 3 × 3 2 = 2 × 3 2 \displaystyle \sqrt{2^{3} \times 3^{2}} = 2 \times 3\, \sqrt{2} 2 3 × 3 2 = 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 \displaystyle 2\, \sqrt{18} = 6\, \sqrt{2} 2 18 = 6 2 WORKED EXAMPLE 1 Simplify 147 \displaystyle \sqrt{147} 147 . Factor 147 as 49 × 3 \displaystyle 49 \times 3 49 × 3 . The square root of 49 is 7. 7 3 \displaystyle 7\, \sqrt{3} 7 3
WORKED EXAMPLE 2 Simplify 25 49 \displaystyle \sqrt{\frac{25}{49}} 49 25 . The numerator and denominator are nonnegative squares. Take their nonnegative roots separately. 5 7 \displaystyle \frac{5}{7} 7 5
Common pitfalls Possible mix-up: a + b \displaystyle \sqrt{a + b} a + b equals a + b \displaystyle \sqrt{a} + \sqrt{b} a + b .
The product rule does not distribute over addition.
Possible mix-up: 48 = 4 3 \displaystyle \sqrt{48} = 4\, \sqrt{3} 48 = 4 3 cannot be checked.
Square 4 3 : 16 × 3 = 48 \displaystyle 4\, \sqrt{3} : 16 \times 3 = 48 4 3 : 16 × 3 = 48 .
Explain it to yourself Why do two different square-factor choices lead to the same simplified radical?
Discuss together Ask the learner to verify a simplified radical by squaring it. Exercises specify which coefficient or radicand to enter.
Preview the eight practice prompts Problem 1
Write 75 = a 3 \displaystyle \sqrt{75} = a\, \sqrt{3} 75 = a 3 with a > 0 \displaystyle a > 0 a > 0 . Find a.
Problem 2
Write 98 = a 2 \displaystyle \sqrt{98} = a\, \sqrt{2} 98 = a 2 with a > 0 \displaystyle a > 0 a > 0 . Find a.
Problem 3
Write 180 = 6 b \displaystyle \sqrt{180} = 6\, \sqrt{b} 180 = 6 b where b is a positive integer. Find b.
Problem 4
Compute 16 81 \displaystyle \sqrt{\frac{16}{81}} 81 16 .
Problem 5
Write 2 45 = a 5 \displaystyle 2\, \sqrt{45} = a\, \sqrt{5} 2 45 = a 5 with a > 0 \displaystyle a > 0 a > 0 . Find a.
Problem 6
Write 288 = a 2 \displaystyle \sqrt{288} = a\, \sqrt{2} 288 = a 2 with a > 0 \displaystyle a > 0 a > 0 . Find a.
Problem 7New context
A square has area 108 square units. Its side is a 3 \displaystyle a\, \sqrt{3} a 3 units with a > 0 \displaystyle a > 0 a > 0 . Find a.
Problem 8New context
A square tile has area 242 cm². Its side is a 2 \displaystyle a\, \sqrt{2} a 2 cm with a > 0 \displaystyle a > 0 a > 0 . What is the integer a?
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.
PARENT EQUATION
Square-root parent: y = x \displaystyle y = \sqrt{x} y = x Starts at (0, 0), curves right and upward, and has real domain x ≥ 0 \displaystyle x \ge 0 x ≥ 0 .
How does the question’s equation change this graph? For y = a x − h + k \displaystyle y = a\, \sqrt{x - h} + k y = a x − h + k , the endpoint is (h, k), the domain is x ≥ h \displaystyle x \ge h x ≥ h , and negative a reflects vertically.
Check: A square root means the nonnegative principal root. Squaring an equation can introduce answers that fail the original equation.
First identify this family. Then read the coefficients and shifts in your actual equation; this reference does not replace the question’s given values.
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 \displaystyle 3\, \sqrt{2} + 5\, \sqrt{2} = 8\, \sqrt{2} 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 \displaystyle \sqrt{12} + \sqrt{27} = 2\, \sqrt{3} + 3\, \sqrt{3} 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 ) = a c b d \displaystyle \left(a\, \sqrt{b}\right)\, \left(c\, \sqrt{d}\right) = ac\, \sqrt{bd} ( a b ) ( c d ) = a c b d WORKED EXAMPLE 1 Simplify 20 + 2 45 \displaystyle \sqrt{20} + 2\, \sqrt{45} 20 + 2 45 . 20 = 2 5 \displaystyle \sqrt{20} = 2\, \sqrt{5} 20 = 2 5 and 2 45 = 6 5 \displaystyle 2\, \sqrt{45} = 6\, \sqrt{5} 2 45 = 6 5 . Combine the coefficients 2 and 6. 8 5 \displaystyle 8\, \sqrt{5} 8 5
WORKED EXAMPLE 2 Compute ( 3 2 ) ( 4 8 ) \displaystyle \left(3\, \sqrt{2}\right)\, \left(4\, \sqrt{8}\right) ( 3 2 ) ( 4 8 ) . Multiply to get 12 16 \displaystyle 12\, \sqrt{16} 12 16 . 16 = 4 \displaystyle \sqrt{16} = 4 16 = 4 . 48
Common pitfalls Possible mix-up: 2 + 3 \displaystyle \sqrt{2} + \sqrt{3} 2 + 3 equals 5 \displaystyle \sqrt{5} 5 .
The product rule does not work for sums.
Possible mix-up: 3 5 + 2 5 \displaystyle 3\, \sqrt{5} + 2\, \sqrt{5} 3 5 + 2 5 equals 5 10 \displaystyle 5\, \sqrt{10} 5 10 .
Keep the shared factor 5 \displaystyle \sqrt{5} 5 : the sum is 5 5 \displaystyle 5\, \sqrt{5} 5 5 .
Explain it to yourself Which operation can make two irrational factors produce a rational result?
Preview the eight practice prompts Problem 1
Write 4 3 + 7 3 = a 3 \displaystyle 4\, \sqrt{3} + 7\, \sqrt{3} = a\, \sqrt{3} 4 3 + 7 3 = a 3 . Find a.
Problem 2
Write 8 + 18 = a 2 \displaystyle \sqrt{8} + \sqrt{18} = a\, \sqrt{2} 8 + 18 = a 2 . Find a.
Problem 3
Compute 6 × 24 \displaystyle \sqrt{6} \times \sqrt{24} 6 × 24 .
Problem 4
Compute ( 2 5 ) ( 3 5 ) \displaystyle \left(2\, \sqrt{5}\right)\, \left(3\, \sqrt{5}\right) ( 2 5 ) ( 3 5 ) .
Problem 5
Write 3 12 − 27 = a 3 \displaystyle 3\, \sqrt{12} - \sqrt{27} = a\, \sqrt{3} 3 12 − 27 = a 3 . Find a.
Problem 6
Compute 75 ÷ 3 \displaystyle \sqrt{75} \div \sqrt{3} 75 ÷ 3 .
Problem 7New context
A rectangle has sides 2 3 \displaystyle 2\, \sqrt{3} 2 3 and 5 3 \displaystyle 5\, \sqrt{3} 5 3 meters. What is its area in square meters?
Problem 8New context
A path has segments 50 \displaystyle \sqrt{50} 50 , 8 \displaystyle \sqrt{8} 8 and 18 \displaystyle \sqrt{18} 18 meters. Its total length is a 2 \displaystyle a\, \sqrt{2} a 2 meters. Find a.
Open stage 10.3 in the student workspace → Browse another level → 
