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Guided Practice

A perfect square is an integer p which has a factor r such that r2 = p.

So, 9 is a perfect square because it has a factor 3 such that 32 = 9.

The square root of any integer that is not a perfect square is an irrational number. For example, 90 is not a perfect square. So, √90 is an irrational number. To the nearest thousandth, √90 ≈ 9.487.

You can use the square root key on a calculator to calculate square roots of numbers.

To estimate the square root of an integer, you can determine between which two consecutive integers the square root falls.

The integer 90 is between the perfect squares of 81 and 100. So, the square root of 90 is between the square root of 81, which is 9, and the square root of 100, which is 10: √90 is between 9 and 10.

The square root of 2, √2, is not a rational
number, meaning that it can't be written as the ratio of two .

When you multiply √2 by itself (√2 × √2), the product is .

Tell whether the number is rational or irrational. If it is rational, write it as a ratio of two integers (e.g., 3/4). If it is irrational, write it as a positive decimal to the nearest thousandth.

The number √25 is a(n) number.

25 =

Tell whether the number is rational or irrational. If it is rational, write it as a ratio of two integers (e.g., 3/4). If it is irrational, write it as a positive decimal to the nearest thousandth.

The number √36 is a(n) number.

36 =

Tell whether the number is rational or irrational. If it is rational, write it as a ratio of two integers (e.g., 3/4). If it is irrational, write it as a positive decimal to the nearest thousandth.

The number √24 is a(n) number.

24 =

Tell whether the number is rational or irrational. If it is rational, write it as a ratio of two integers (e.g., 3/4). If it is irrational, write it as a positive decimal to the nearest thousandth.

The number √121 is a(n) number.

121 =

Tell whether the number is rational or irrational. If it is rational, write it as a ratio of two integers (e.g., 3/4). If it is irrational, write it as a positive decimal to the nearest thousandth.

The number √48 is a(n) number.

48 =

Estimate the value of the irrational number by identifying the consecutive integers it falls between.

22 is between and .

Estimate the value of the irrational number by identifying the consecutive integers it falls between.

54 is between and .

Estimate the value of the irrational number by identifying the consecutive integers it falls between.

10 is between and .

Estimate the value of the irrational number by identifying the consecutive integers it falls between.

40 is between and .

Estimate the value of the irrational number by identifying the consecutive integers it falls between.

130 is between and .

Enter the name of the point on the number line which best represents the location of √88.

Point represents the location of √88.

Enter the name of the point on the number line which best represents the location of √5.

Point represents the location of √5.

Enter the name of the point on the number line which best represents the location of √35.

Point represents the location of √35.

The square root of 1 is  .

The square root of 0 is .

The Pythagorean Theorem says that, in a right triangle, a2 + b2 = c2, where c represents the hypotenuse length.

This means that the hypotenuse length, c, is equal to √a2 + b2.

If a = 3 and b = 4, then c = √32 + 42, which is equal to √25, or 5.

You can use square roots to determine the side length of a square given its area. For example, suppose the area of this square, with side length s, is 62 square inches. What is the length of each side, s?

The area is equal to s2, so each side length is √62 in., or approximately 7.874 in.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the length of the hypotenuse, c. If the hypotenuse length is an irrational number, round to the nearest thousandth.

The hypotenuse length, c, is units.

Determine the side length, s, of the square. If the length is an irrational number, round to the nearest thousandth.

A square has an area of 225 square units. What is the length of each side of the square, s?

Each side is unit(s) long.

Determine the side length, s, of the square. If the length is an irrational number, round to the nearest thousandth.

A square has an area of 72 square units. What is the length of each side of the square, s?

Each side is unit(s) long.

Determine the side length, s, of the square. If the length is an irrational number, round to the nearest thousandth.

A square has an area of 18 square units. What is the length of each side of the square, s?

Each side is unit(s) long.

You can take the square roots of fractions and decimals too.

The shaded area shows 4/25. And it shows that the square root of 4/25, √4/25, is 2/5. Notice that √4/25 = √4/√25.

You can compare square roots also. If a < b, then √a < √b.

Use the example to determine this square root. Write your answer as a decimal.

0.16 =

Write >, <, or = to compare the square roots.

9/16 4/81

Determine the square root. Write your answer as a fraction (e.g., 1/2).

9/64 =

Determine the square root. Write your answer as a fraction (e.g., 1/2).

49/4 =

Write >, <, or = to compare the square roots.

0.25 3/12

Determine the square roots. Write your answers as integers.

100 = ,00

10,000 =

Write >, <, or = to compare the square roots.

0.0121 0.0144

Determine the square root. Write your answer as a fraction (e.g., 1/2).

25/100 =

Determine the square root. Write your answer as a fraction (e.g., 1/2).

1/9 =

Determine the square root. Write your answer as a decimal.

0.36 =

Write >, <, or = to compare the square roots.

 1  81/100

Determine the square root. Write your answer as a fraction (e.g., 1/2).

25/49 =

0.00

The number line shows the locations, in red, of different square roots—from the square root of 2 to the square root of 20.

We skipped over the square roots of 4, 9, and 16, since those are perfect squares.

Move your mouse to position the right triangle and then click to draw a circle. Estimate the position you should put the cursor so that the circle intersects each square root.

Explain how you figured it out.

You know that a rational number is a number which can be written as the ratio of two integers. There are some numbers that can't be written as the ratio of two integers. These are called irrational numbers.

Any square root of an odd number (except for the square root of 1) or prime number is an irrational number. But some square roots of even numbers, like √6 are also irrational numbers.

The number π is a famous irrational number, approximately equal to 3.14159.

Write rational or irrational to describe each number.

2 is .

4 is .

3 is .

5 is .