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Algebra I

Square Roots and Radicals

Lesson

The square root of a number is what you multiply by itself to get that number. 25=5\sqrt{25} = 5 because 5⋅5=255 \cdot 5 = 25.

Perfect squares are numbers whose square root is a whole number — memorizing them helps:

1,4,9,16,25,36,49,64,81,100,121,1441, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Simplifying radicals

For numbers that aren’t perfect squares, we simplify by pulling out the largest perfect-square factor:

a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

Find the largest perfect square that divides your number. Take its square root and bring it outside; leave the rest under the radical.

Worked example 1

50\sqrt{50}

50 = 25 × 2, and 25 is a perfect square. So:

50=25⋅2=25⋅2=52\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}

Worked example 2

72\sqrt{72}

72 = 36 × 2, and 36 is a perfect square (the largest one that divides 72):

72=36⋅2=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}

How to type your answer

Use sqrt(n) for the square root of nn. For a coefficient times a radical, write the coefficient first with no space: 5sqrt(2), 6sqrt(2), 2sqrt(7). If the answer is a whole number, just type the number: 5.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

4\sqrt{4}

Problem 2

9\sqrt{9}

Problem 3

49\sqrt{49}

Problem 4

100\sqrt{100}

Practice

Standard problems matching the lesson.

Problem 5

A square plot has area 25 sq ft. Side length?

Problem 6

A square room has area 81 sq m. Side length?

Problem 7

A square garden has area 144 sq ft. Side length?

Problem 8

8\sqrt{8}

Problem 9

12\sqrt{12}

Problem 10

18\sqrt{18}

Problem 11

20\sqrt{20}

Problem 12

27\sqrt{27}

Problem 13

50\sqrt{50}

Problem 14

45\sqrt{45}

Problem 15

72\sqrt{72}

Problem 16

32\sqrt{32}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

75\sqrt{75}

Problem 18

98\sqrt{98}

Problem 19

147\sqrt{147}

Problem 20

200\sqrt{200}

Problem 21

48\sqrt{48}

Problem 22

300\sqrt{300}

Practice

Standard problems matching the lesson.

Problem 23

A square room has area 50 square feet. Find its side length in simplified radical form (ft).

Problem 24

The diagonal of a 6-by-6 square satisfies d^2 = 72. Find d in simplified radical form.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A square garden has area 200 square meters. Find its side length in simplified radical form (m).

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