Simplifying Radicals
A square root is in simplest form when no perfect square is left under the sign. Then you can add and compare roots.
What you'll be able to do
- Write a square root or cube root in simplest radical form.
- Add and subtract radicals after you simplify them.
- Multiply radicals, and clear a root from the bottom of a fraction.
- Give exact lengths with radicals.
The big idea
Split off a perfect square. √(ab) = √a · √b. So √63 = √(9 · 7) = 3√7.
Take the biggest one. For √200, use 100, not 4. √200 = √(100 · 2) = 10√2.
Cube roots use perfect cubes. ∛81 = ∛(27 · 3) = 3∛3.
Letters work the same way. Pull out even powers: √(x⁷) = √(x⁶ · x) = x³√x.
Add only like radicals. 2√3 + 5√3 = 7√3, the same way 2 apples + 5 apples = 7 apples. But √2 + √3 stays as it is.
Multiply inside the root. √6 · √10 = √60 = √(4 · 15) = 2√15.
No root on the bottom. Multiply the top and bottom by the root: 10/√5 = 10√5/5 = 2√5.
Compare by moving numbers inside. 3√2 = √18 and 2√5 = √20. So 2√5 is bigger.
Words you need
- Radical
- A root sign and what is under it, like √5 or ∛x.
- Radicand
- The number under the root sign. In √18, it is 18.
- Square root
- A number that gives the first number when you multiply it by itself. √25 = 5.
- Cube root
- A number you use 3 times as a factor to get the first number. ∛8 = 2.
- Perfect square
- A whole number times itself, like 4, 9, 16, 25 and 36.
- Perfect cube
- A whole number used 3 times as a factor, like 8, 27, 64 and 125.
- Simplest radical form
- No perfect square is left under a square root, and no root is left on the bottom of a fraction.
- Simplify
- Write it in a shorter form with the same value.
- Like radicals
- Radicals with the same number under the sign, like 2√3 and 5√3.
- Factor
- A number you multiply by. 9 and 7 are factors of 63.
- Product
- The answer when you multiply.
- Quotient
- The answer when you divide.
- Coefficient
- The number in front. In 5√3, it is 5.
- Rationalize
- Rewrite a fraction so the bottom has no root.
- Numerator
- The top number of a fraction.
- Denominator
- The bottom number of a fraction.
- Hypotenuse
- The longest side of a right triangle. It is across from the right angle.
- Base
- The bottom of a shape, like the bottom of a box. In a power like 3², the base is 3.
- Variable
- A letter that stands for a number, like x.
- Expression
- Numbers and letters joined by +, −, × or ÷. It has no = sign.
- Exponent
- The small raised number. In x⁷, it is 7.
- Power
- A base with an exponent, like 3².
- Logarithm
- The power a base needs to make a number. log₃(9) = 2, because 3² = 9.
Perfect squares and cubes to look for
| Perfect squares | Perfect cubes |
|---|---|
| 4, 9, 16, 25 | 8, 27 |
| 36, 49, 64, 81 | 64, 125 |
| 100, 121, 144 | 216, 1,000 |
Worked examples
Example 1
Write √180 in simplest radical form.
- Find perfect squares that divide 180: 4, 9 and 36.
- The biggest is 36. So √180 = √(36 · 5).
- √36 = 6. So √180 = 6√5.
Answer: 6√5
Example 2
Simplify √28 + √63 − √7.
- Simplify each root first: √28 = √(4 · 7) = 2√7 and √63 = √(9 · 7) = 3√7.
- Now all three are like radicals: 2√7 + 3√7 − 1√7.
- Add and subtract the numbers in front: 2 + 3 − 1 = 4. The answer is 4√7.
Answer: 4√7
Example 3
A tent pole is 6 feet tall. A rope goes from the top of the pole to a stake 2 feet from its base. How long is the rope? Give an exact answer in simplest radical form.
- The pole, the ground and the rope make a right triangle. The rope is the hypotenuse.
- c² = 6² + 2² = 36 + 4 = 40.
- c = √40 = √(4 · 10) = 2√10 feet.
Answer: 2√10 feet
Example 4
Write 6/√12 in simplest radical form.
- Simplify the bottom first: √12 = 2√3. So 6/√12 = 6/(2√3) = 3/√3.
- Multiply the top and bottom by √3: 3√3/(√3 · √3) = 3√3/3.
- Divide 3 by 3. The answer is √3.
Answer: √3
Common mistakes
Splitting a root over a sum. √(36 + 64) is √100 = 10, not 6 + 8.
Adding the numbers inside. √2 + √8 is not √10. Simplify first: √2 + 2√2 = 3√2.
Stopping too soon. √48 = 2√12 is not done, because 12 still holds a 4. √48 = 4√3.
Moving a number inside without squaring it. 3√2 is √18, not √6.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Simplify √98 and ∛128. Give both answers in simplest radical form.
- Zanele designs a square solar panel with an area of 75 square feet. Find the exact length of one side of the panel in feet, and find its exact perimeter in feet. Give both answers in simplest radical form. Word problem
- A square patio has an area of 72 square meters. Cormac finds the side length by writing √72 = √(4 · 18) = 2√18 meters and stops there. Explain why his answer is not in simplest radical form, and give the correct side length in simplest radical form with units. Word problem
- Two steel cables brace a radio tower. Cable A is 3√5 meters long and Cable B is 2√11 meters long. Without a calculator, decide which cable is longer, and show how you know. Word problem
- Simplify √(50a⁵b²). Give your answer in simplest radical form. All variables are positive.
Tier 2 Intermediate One more step to think through
- Satsuki builds a wooden frame around a rectangular painting whose width is √20 inches and whose length is √45 inches. What is the exact perimeter of the painting in inches? Give the answer in simplest radical form. Word problem
- Arash simplifies √12 + √75 − √3 by adding and subtracting the numbers under the radical signs, getting √84 = 2√21. Explain his mistake and find the correct answer in simplest radical form.
- Simplify √6 · √15 and √3(√12 − 5). Give both answers in simplest radical form.
- Milena pours 12 cubic inches of melted wax into a prism-shaped mold with a base area of √6 square inches. The height of the wax equals the volume divided by the base area. Find the exact height of the wax in inches. Give your answer in simplest radical form with a rationalized denominator. Word problem
- Jengo compares the diagonals of two rectangular garden beds. Bed A measures 4 meters wide by 6 meters long. Bed B has a diagonal of 3√6 meters. Without a calculator, find the exact diagonal of Bed A in simplest radical form and determine which bed has the longer diagonal. Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A 10-foot ladder leans against a vertical wall. The base of the ladder sits 2√7 feet from the bottom of the wall on level ground. How high up the wall does the top of the ladder reach? Give the exact height in feet in simplest radical form. Word problem
- Simplify ∛(54x⁷y³) − x²y∛(16x). Give the answer in simplest radical form. All variables are positive.
- Paloma cuts a rectangular sheet of copper whose area is 30 square centimeters and whose width is √15 centimeters. What is the exact length of the sheet in centimeters, and what is its exact perimeter in centimeters? Give both answers in simplest radical form. Word problem
- Ruslan walks 3 kilometers east and 4 kilometers north to reach a trailhead. Then he walks another √18 kilometers along a straight trail. To find his straight-line distance from the start to the trailhead, he writes √(3² + 4²) = √9 + √16 = 3 + 4 = 7 kilometers. Find his mistake and give the correct distance to the trailhead in kilometers. Then write his total distance walked, the two legs plus the trail, in simplest radical form. Word problem
- Number A is 18 / √3. Number B is √27 + √12. Simplify both numbers to simplest radical form. Which number is greater, and by how much?
Tier 4 Challenge Stretch problems, some above grade level
- Ingrid places three square tiles side by side in a row along a straight edge. The areas of the three tiles are 18 square inches, 50 square inches, and 98 square inches. What is the exact total length of the bottom edge formed by all three tiles in inches, in simplest radical form? What would be the area, in square inches, of a single large square whose side equals that total length? Word problem
- Benoit simplifies 10a / √(5a³) for a positive number a. He multiplies the top and bottom by √(5a³), writes 10a√(5a³) / (5a³) = 2√(5a³) / a², and stops. Explain why his answer is not in simplest radical form and find the completely simplified expression.
- Leilani builds a stage prop from two right triangles. Triangle A has legs of length √10 meters and √14 meters. Triangle B has a hypotenuse of 5 meters and one leg of 1 meter. Find the exact hypotenuse of Triangle A and the exact unknown leg of Triangle B in simplest radical form, and state how the two lengths compare. Word problem
- Orhan cuts a wooden trim board of length √147 centimeters into two pieces with no waste. One piece has length 4√3 centimeters, and the other piece has length √k centimeters. Find the exact whole number k. Word problem
- A logarithm tells you a power. log₃(9) = 2 because 3² = 9. Find the exact value of S = log₂(√32 · √8) + log₃(9 / √27) as a fraction. Above grade level
Answers
Tier 1 Foundations
- √98 = 7√2 and ∛128 = 4∛2. Factor out the largest perfect square from 98: √98 = √(49 · 2) = √49 · √2 = 7√2. Factor out the largest perfect cube from 128: ∛128 = ∛(64 · 2) = ∛64 · ∛2 = 4∛2.
- Side length is 5√3 feet; perimeter is 20√3 feet. The side length of a square with area 75 is √75. Factor out the largest perfect square: √75 = √(25 · 3) = 5√3 feet. Multiply the side length by 4 to find the perimeter: 4 · 5√3 = 20√3 feet.
- The number 18 inside the radical still has a perfect-square factor of 9; the correct side length is 6√2 meters. Check the radicand 18: since 18 = 9 · 2 and 9 is a perfect square, 2√18 is not completely simplified. Pull out √9 = 3 from √18: 2√18 = 2 · 3√2 = 6√2 meters. You can also factor out the largest perfect square 36 in one step: √72 = √(36 · 2) = 6√2 meters.
- Cable A is √45 meters and Cable B is √44 meters; Cable A is longer. Square the outside factor 3 to move it inside the radical for Cable A: 3√5 = √(9 · 5) = √45 meters. Square the outside factor 2 to move it inside the radical for Cable B: 2√11 = √(4 · 11) = √44 meters. Since 45 > 44, √45 > √44, so Cable A is longer.
- 5a²b√(2a). Split each factor into the largest perfect square times the leftover factor: 50 = 25 · 2, a⁵ = a⁴ · a, and b² is already a perfect square. Take the square root of the perfect squares: √25 = 5, √(a⁴) = a², and √(b²) = b. Leave the remaining factors 2 and a inside the radical: 5a²b√(2a).
Tier 2 Intermediate
- 10√5 inches. Simplify each dimension first: √20 = √(4 · 5) = 2√5 inches and √45 = √(9 · 5) = 3√5 inches. Add the length and width by combining like radicals: 2√5 + 3√5 = 5√5 inches. Double that sum for the perimeter: 2 · 5√5 = 10√5 inches.
- Arash combined the numbers under separate square roots, but you can only add radicals with the same number inside; the correct answer is 6√3. Radicals can only be added or subtracted when they have the exact same radicand. Simplify each term first: √12 = √(4 · 3) = 2√3 and √75 = √(25 · 3) = 5√3. Combine the coefficients of the like √3 terms: 2√3 + 5√3 − 1√3 = 6√3.
- 3√10 and 6 − 5√3. Multiply the radicands in the first expression and simplify: √6 · √15 = √90 = √(9 · 10) = 3√10. Distribute √3 across both terms in the second expression: √3 · √12 − √3 · 5 = √36 − 5√3. Simplify √36 to 6, giving 6 − 5√3.
- 2√6 inches. Write the height as the quotient 12 / √6. Multiply the numerator and denominator by √6 to rationalize the denominator: (12 · √6) / (√6 · √6) = 12√6 / 6. Divide the outside coefficients: 12 ÷ 6 = 2, giving 2√6 inches.
- Bed A's diagonal is 2√13 meters; Bed B has the longer diagonal. Use the Pythagorean theorem for Bed A's diagonal: d = √(4² + 6²) = √(16 + 36) = √52. Simplify √52: √(4 · 13) = 2√13 meters. Compare √52 with Bed B's diagonal written as a single root: 3√6 = √(9 · 6) = √54. Since √54 > √52, Bed B has the longer diagonal.
Tier 3 Advanced
- 6√2 feet. Set up the Pythagorean theorem with height h and hypotenuse 10: h² + (2√7)² = 10². Square 2√7 by squaring both factors: (2√7)² = 4 × 7 = 28, so h² + 28 = 100. Subtract 28 to get h² = 72, and simplify: h = √72 = √(36 · 2) = 6√2 feet.
- x²y∛(2x). In the first term, factor out the largest perfect cubes: 54 = 27 · 2, x⁷ = x⁶ · x, and y³ is a perfect cube. Taking cube roots gives 3x²y∛(2x). In the second term, factor 16 = 8 · 2 and take the cube root of 8: x²y · 2∛(2x) = 2x²y∛(2x). Subtract the like radical terms: 3x²y∛(2x) − 2x²y∛(2x) = x²y∛(2x).
- Length is 2√15 centimeters; perimeter is 6√15 centimeters. Divide area by width to find the length: 30 / √15. Rationalize the denominator: (30√15) / 15 = 2√15 centimeters. Find the perimeter: 2(√15 + 2√15) = 2(3√15) = 6√15 centimeters.
- You cannot split a square root across a sum; the straight-line distance to the trailhead is 5 kilometers, and his total distance walked is 7 + 3√2 kilometers. Add inside the radical before taking the square root: √(9 + 16) = √25 = 5 kilometers, not √9 + √16. Simplify the trail distance: √18 = √(9 · 2) = 3√2 kilometers. Add the walking segments: 3 + 4 + 3√2 = 7 + 3√2 kilometers.
- A is 6√3 and B is 5√3; A is greater by √3. Rationalize Number A: 18 / √3 = (18√3) / 3 = 6√3. Simplify and add Number B: √27 + √12 = 3√3 + 2√3 = 5√3. Subtract: 6√3 − 5√3 = √3.
Tier 4 Challenge
- Total length is 15√2 inches; area of the large square is 450 square inches. Find each tile's side length by simplifying its square root: √18 = 3√2 inches, √50 = 5√2 inches, and √98 = 7√2 inches. Add the three like radicals to find the total length: 3√2 + 5√2 + 7√2 = 15√2 inches. Square 15√2 to find the large square's area: (15√2)² = 15² × (√2)² = 225 × 2 = 450 square inches.
- The radical √(5a³) still contains the perfect square factor a²; pulling out a and canceling gives 2√(5a) / a. Inside √(5a³), factor a³ as a² · a and take the square root of a² to get a√(5a). Substitute this into Benoit's fraction: 2a√(5a) / a². Cancel one factor of a from the numerator and denominator: 2√(5a) / a.
- Both lengths are 2√6 meters, so they are equal. For Triangle A, square both radical legs and add: c = √((√10)² + (√14)²) = √(10 + 14) = √24 = 2√6 meters. For Triangle B, subtract the known leg squared from the hypotenuse squared: b = √(5² − 1²) = √(25 − 1) = √24 = 2√6 meters. Both lengths simplify to 2√6 meters, so they are equal.
- k = 27. Simplify the original board length: √147 = √(49 · 3) = 7√3 centimeters. Subtract the first piece to find the second piece's length: 7√3 − 4√3 = 3√3 centimeters. Rewrite 3√3 as a single square root by squaring the outside 3 and multiplying inside: 3√3 = √(9 · 3) = √27, so k = 27.
- S = 9/2. Simplify the first radical product: √32 · √8 = √256 = 16. Since 2⁴ = 16, log₂(16) = 4. Simplify the second radical quotient: 9 / √27 = 9 / (3√3) = 3 / √3 = √3. Since √3 = 3¹ᐟ², log₃(√3) = 1/2. Add the two parts: S = 4 + 1/2 = 9/2.
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