How can you approximate the square root of a number that is not a perfect square?

When a number is not a perfect square, its square root cannot be expressed as a whole number or simple fraction. To approximate it, you can use two common methods. First, use a calculator with a square root function to get a decimal approximation. For example, the square root of 27 is approximately 27 ≈ 5.196. Second, use a number line by identifying perfect squares closest to the number. Since 27 lies between 25 and 36, whose roots are 5 and 6 respectively, the square root of 27 is about 5.1, closer to 5. This method helps visualize and estimate square roots without a calculator.

9. Geometry

Square Roots and The Pythagorean Theorem

9. Geometry

Square Roots and The Pythagorean Theorem: Videos & Practice Problems

Video Lessons

Topic summary

Square Roots and The Pythagorean Theorem connect the idea of reversing a square with finding missing side lengths in a right triangle. A square root asks what number multiplied by itself gives a value, and the radical sign represents the principal square root, meaning the positive answer. Students work with perfect squares, including some fractions, and learn that when a number is not a perfect square, its square root can be approximated with a calculator or located between nearby perfect squares on a number line.

The Pythagorean theorem applies only to right triangles and relates the side lengths by $a^2+b^2=c^2$ . The legs are the two shorter sides, and the hypotenuse is the longest side opposite the right angle. To find an unknown side, substitute the known lengths, solve for the squared value, and then take a square root; answers may be exact radicals or decimal approximations.

Concept

Introduction to Square Roots

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Problem

Evaluate.
(A) $the square root of 225$

$5$

$15$

$25$

$9$

Problem

Evaluate.
(B) $\sqrt{\frac{4}{121}}$

$2 over 121$

$4 over 11$

$2 over 11$

$4 over 121$

Problem

Evaluate.
(C) $\sqrt{\frac{25}{64}}$

$\frac{5}{8}$

$5 over 64$

$25 over 8$

$25 over 64$

Concept

Approximating Square Roots

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Example

Approximating Square Roots Example 1

Video duration:

49s

Problem

Determine what two whole numbers the square root is between without using a calculator.
(A) $the square root of 58$

$7$ and $8$

$6$ and $7$

$7$ and $9$

$8$ and $9$

Problem

Determine what two whole numbers the square root is between without using a calculator.
(B) $the square root of 175$

$13$ and $15$

$13$ and $14$

$14$ and $15$

$12$ and 14

Problem

Approximate using a calculator and round to the nearest thousandth.
(A) $the square root of 19$

$4.242$

$4.243$

$4.358$

$4.359$

Problem

Approximate using a calculator and round to the nearest thousandth.
(B) $the square root of 51$

$7.141$

$7.142$

$7.211$

$7.212$

Concept

Solving Right Triangles with the Pythagorean Theorem

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Solving Right Triangles with the Pythagorean Theorem Video Summary

In the study of right triangles, a fundamental concept is the Pythagorean theorem, which is essential for solving problems involving the lengths of the sides. This theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (often referred to as legs) is equal to the square of the length of the longest side, known as the hypotenuse. The equation can be expressed as:

\[a^2 + b^2 = c^2\]

Here, $a$ and $b$ represent the lengths of the legs, while $c$ represents the length of the hypotenuse. It is crucial to remember that this theorem applies only to right triangles, where one angle measures 90 degrees.

To illustrate the application of the Pythagorean theorem, consider a right triangle where one leg measures 4 units and the other leg measures 3 units. To find the hypotenuse $c$, we can substitute the values into the equation:

\[4^2 + 3^2 = c^2\]

Calculating this gives:

\[16 + 9 = c^2\]

Thus, we have:

\[25 = c^2\]

To find $c$, we take the square root of both sides:

\[c = \sqrt{25} = 5\]

This indicates that the hypotenuse of the triangle is 5 units long, which is consistent with the property that the hypotenuse is the longest side.

In another scenario, if we know the hypotenuse and one leg, we can still use the Pythagorean theorem to find the unknown leg. For example, if the hypotenuse $c$ is 10 units and one leg $b$ is 6 units, we can set up the equation as follows:

\[a^2 + 6^2 = 10^2\]

Substituting the known values gives:

\[a^2 + 36 = 100\]

To isolate $a^2$, we subtract 36 from both sides:

\[a^2 = 100 - 36\]

Thus, we find:

\[a^2 = 64\]

Taking the square root of both sides yields:

\[a = \sqrt{64} = 8\]

In this case, the length of the unknown leg is 8 units, which again confirms that the hypotenuse (10 units) is longer than both legs.

Understanding and applying the Pythagorean theorem is crucial for solving various geometric problems involving right triangles, making it a key concept in mathematics.

Problem

Calculate the missing side of the triangle below.

$25$

$26$

$23$

$31$

Problem

Calculate the missing side of the triangle below.

Problem

Calculate the missing side of the triangle below.

$2$\sqrt$5$

$2$

$5$

$2$\sqrt$3$

Example

Right Triangles & Pythagorean Theorem Example 2

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Here's what students ask on this topic:

The principal square root of a number is the positive value that, when multiplied by itself, equals the original number. It is represented by the radical symbol $\sqrt{x}$ . For example, the principal square root of 25 is 5 because $\sqrt{25} = 5$ . Although -5 multiplied by itself also equals 25, the radical symbol alone refers only to the positive root. To indicate the negative root, a negative sign is placed before the radical, like $- \sqrt{25} = - 5$ . In most math courses, including prealgebra, we focus on the principal (positive) square root to keep answers consistent and clear.

When a number is not a perfect square, its square root cannot be expressed as a whole number or simple fraction. To approximate it, you can use two common methods. First, use a calculator with a square root function to get a decimal approximation. For example, the square root of 27 is approximately $\sqrt{27} ≈ 5.196$ . Second, use a number line by identifying perfect squares closest to the number. Since 27 lies between 25 and 36, whose roots are 5 and 6 respectively, the square root of 27 is about 5.1, closer to 5. This method helps visualize and estimate square roots without a calculator.

The Pythagorean theorem relates the sides of a right triangle with the formula $a^{2} + b^{2} = c^{2}$ , where $a$ and $b$ are the legs (shorter sides) and $c$ is the hypotenuse (longest side). To find a missing side, substitute the known side lengths into the equation. For example, if you know the legs are 3 and 4, then $3^{2} + 4^{2} = x^{2}$ gives $9 + 16 = x^{2}$ , so $x^{2} = 25$ . Taking the square root, $x = \sqrt{25} = 5$ . This method works for any missing side as long as the triangle is right-angled.

Perfect squares are numbers that are the square of an integer or fraction. For example, 9, 16, and 25 are perfect squares because they equal $3^{2}, 4^{2}, and 5^{2}$ respectively. Their square roots are whole numbers or simple fractions. For instance, the square root of 144 is 12 because $12^{2} = 144$ . Recognizing perfect squares helps simplify square root calculations and identify when exact answers are possible instead of approximations.

To find the square root of a fraction, take the square root of the numerator and the denominator separately. For example, to find the square root of $\frac{1}{81}$ , find the square root of 1 and 81 individually. The square root of 1 is 1, and the square root of 81 is 9, so the square root of the fraction is $\frac{1}{9}$ . This method works because the square root operation distributes over division for positive numbers.

Your Prealgebra tutors

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Square Roots and The Pythagorean Theorem: Videos & Practice Problems

Introduction to Square Roots

Evaluate.
(A) $the square root of 225$

Evaluate.
(B) $\sqrt{\frac{4}{121}}$

Evaluate.
(C) $\sqrt{\frac{25}{64}}$

Approximating Square Roots

Approximating Square Roots Example 1

Determine what two whole numbers the square root is between without using a calculator.
(A) $the square root of 58$

Determine what two whole numbers the square root is between without using a calculator.
(B) $the square root of 175$

Approximate using a calculator and round to the nearest thousandth.
(A) $the square root of 19$

Approximate using a calculator and round to the nearest thousandth.
(B) $the square root of 51$

Solving Right Triangles with the Pythagorean Theorem

Solving Right Triangles with the Pythagorean Theorem Video Summary

Calculate the missing side of the triangle below.

Calculate the missing side of the triangle below.

Calculate the missing side of the triangle below.

Right Triangles & Pythagorean Theorem Example 2

Here's what students ask on this topic:

What is the principal square root and how is it different from negative square roots?

How can you approximate the square root of a number that is not a perfect square?

How do you use the Pythagorean theorem to find a missing side of a right triangle?

What are perfect squares and how do they relate to square roots?

How do you find the square root of a fraction?

Your Prealgebra tutors

Square Roots and The Pythagorean Theorem: Videos & Practice Problems

Introduction to Square Roots

Evaluate.(A) 225\(\sqrt{225}\)

Evaluate.(B) 4121\(\sqrt{\frac{4}{121}\)}

Evaluate.(C) 2564\(\sqrt{\frac{25}{64}\)}

Approximating Square Roots

Approximating Square Roots Example 1

Determine what two whole numbers the square root is between without using a calculator.(A) 58\(\sqrt{58}\)

Determine what two whole numbers the square root is between without using a calculator.(B) 175\(\sqrt{175}\)

Approximate using a calculator and round to the nearest thousandth.(A) 19\(\sqrt{19}\)

Approximate using a calculator and round to the nearest thousandth.(B) 51\(\sqrt{51}\)

Solving Right Triangles with the Pythagorean Theorem

Solving Right Triangles with the Pythagorean Theorem Video Summary

Calculate the missing side of the triangle below.

Calculate the missing side of the triangle below.

Calculate the missing side of the triangle below.

Right Triangles & Pythagorean Theorem Example 2

Here's what students ask on this topic:

What is the principal square root and how is it different from negative square roots?

What is the principal square root and how is it different from negative square roots?

How can you approximate the square root of a number that is not a perfect square?

How can you approximate the square root of a number that is not a perfect square?

How do you use the Pythagorean theorem to find a missing side of a right triangle?

How do you use the Pythagorean theorem to find a missing side of a right triangle?

What are perfect squares and how do they relate to square roots?

What are perfect squares and how do they relate to square roots?

How do you find the square root of a fraction?

How do you find the square root of a fraction?

Your Prealgebra tutors

Evaluate.
(A) $the square root of 225$

Evaluate.
(B) $\sqrt{\frac{4}{121}}$

Evaluate.
(C) $\sqrt{\frac{25}{64}}$

Determine what two whole numbers the square root is between without using a calculator.
(A) $the square root of 58$

Determine what two whole numbers the square root is between without using a calculator.
(B) $the square root of 175$

Approximate using a calculator and round to the nearest thousandth.
(A) $the square root of 19$

Approximate using a calculator and round to the nearest thousandth.
(B) $the square root of 51$