Topic 2 of 8

Square roots and irrational numbers

Step 10 · Pre-algebra topics

Find square and cube roots, solve x² = p and x³ = p, tell rational from irrational numbers, turn repeating decimals into fractions, and estimate roots.

Square roots and cube roots

The square root of a number is what you square to get it: √49 is 7 because 7² = 49. A square with an area of 49 has sides of 7.

A square of 49 unit squares has sides of 7, so √49 = 7.

The cube root undoes cubing: the cube root of 64 is 4, because 4³ = 64. A cube made of 64 unit cubes has edges of 4.

A cube of 64 unit cubes has edges of 4: the cube root of 64 is 4.

Example: Solving x² = p

Solve x² = 81.

9² = 81, and (−9)² = 81 too.
So there are two answers.

Answerx = 9 or x = −9

Example: Solving x³ = p

Solve x³ = 125.

5³ = 125.
Only one number cubes to 125.

Answerx = 5

Rational and irrational numbers

A rational number can be written as a fraction of two integers: 0.75 is $\frac{3}{4}$ , −2 is $\frac{-2}{1}$ , and 0.333… is $\frac{1}{3}$ . Its decimal either stops or repeats.

An irrational number cannot be written as a fraction. Its decimal goes on forever without repeating. √2, √3 and π are irrational.

Number	Rational?	Why
√16	Yes	it is 4
0.125	Yes	it is $\frac{1}{8}$
0.454545…	Yes	it repeats: it is $\frac{5}{11}$
√5	No	5 is not a perfect square
π	No	it never ends or repeats

Example: A repeating decimal as a fraction

Write 0.777… as a fraction.

Call it x. Then 10x is 7.777…
Take x away from 10x: the endless 7s cancel, leaving 9x = 7.
So x is $\frac{7}{9}$ .

Answer $\frac{7}{9}$

Estimating roots

To estimate a square root, find the perfect squares on either side of the number.

Example: Between two whole numbers

Between which two whole numbers is √50?

49 < 50 < 64.
So √50 is between √49 and √64.

AnswerBetween 7 and 8, very close to 7

√50 is just past 7, because 50 is just past 49.

Example: To one decimal place

Estimate √30 to one decimal place.

5² = 25 and 6² = 36, so √30 is between 5 and 6.
5.4² = 29.16 and 5.5² = 30.25.
30 is closer to 30.25.

AnswerAbout 5.5

Example: Comparing

Which is bigger: √20 or 4.5?

4.5² = 20.25.
20 is less than 20.25, so √20 is less than 4.5.

Answer4.5

Try it yourself

Work each one on paper first, then open the answer to check.

What is √144?
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12
What is the cube root of 27?
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3
Solve x² = 64.
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x = 8 or x = −8
Is 0.121212… rational or irrational?
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Rational: it repeats. It is $\frac{12}{99}$ , which is $\frac{4}{33}$ .
Is √12 rational or irrational?
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Irrational: 12 is not a perfect square.
Between which two whole numbers is √90?
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9 and 10, because 81 < 90 < 100.
Put in order, smallest first: √10, 3.5, π.
Show answer
π, √10, 3.5 (π is about 3.14 and √10 is about 3.16).