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Chapter 5—Powers, Exponents, and Roots

5-2 Square Roots and Cube Roots

 When you complete the work for this section, you should be able to: Describe the meaning of the square root of a number. Cite the square roots for perfect-square values between 1 and 100. Demonstrate how to use a calculator to find the square root of any positive number. Describe the meaning of the cube root of a number. Demonstrate how to use a calculator to find the cube root of any number.

Introducing Square Roots

Question: What number multiplied by itself is equal to 4?
Well, 2. x 2 = 4, so the answer is 2.

Question: What number multiplied by itself is equal to 16?
The answer is 4. Why? Because 4 multiplied by itself equals 16.

Those are examples of finding square roots. The square root of 4 is 2. The square root of 16 is 4.

 Definition The square root is a number, when multiplied by itself, is equal to a given number.

The square-root operation is the opposite of the squaring operation:

• The square of 3 is 9 (32 = 9)
• The square root of 9 is 3

If you can find the square of a number, you should be able to determine the square root of the result.

The symbol for the square root of a number is the radical sign, Ö     . Examples

1.   Since 122 = 144, then Ö 144 = 12
2.   Since 252 = 625, then Ö 625 = 25
3,   Since 1002 = 10000, then Ö 10000 = 100
 Squares Square Roots 12 = 1 Ö 1 = 1 22 = 4 Ö 4 = 2 32 = 9 Ö 9 = 3 42 = 16 Ö 16 = 4 52 = 25 Ö 25 = 5 62 = 36 Ö 36 = 6 72 = 49 Ö 49 = 7 82 = 64 Ö 64 = 8 92 = 81 Ö 81 = 9 102 = 100 Ö 100 = 10

This table  lists the Squares for integers between 1 and 9. The Square Roots column shows how we can use square roots to convert the squares back to their roots.

Examples & Exercises

 Squares and Square Roots Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

 Notes About Square Roots The square root of zero is zero: Ö 0 = 0 Don't bother trying to find the square root of a negative number. The solution exists, but not in the real number system. Pre-algebra courses deal only with the real number system, so you aren't responsible for finding square roots of negative numbers.

Finding the Square Root of Any Real Number

So far in this lesson, you've been finding the square root of a particular set of integers called perfect squares. A perfect square is an integer that is found by squaring another integer.  You already know how to find the square root of 25 because it is a perfect square: 5 x 5 = 25, or you could write it as 52 = 25. So 25 is a perfect square, and its square root is 5. Not all numbers, however, are perfect squares--and not all square roots are so easy to find.

What is the square root of 12 ( Ö 12 )?
The square root of 12 is somewhere between 3 and 4, because 32 = 9 and 42 = 16.
So the square root of 12 is a decimal fraction somewhere between 3 and 4.
How do we find the solution?

Back in the "old days," math students had to learn how to calculate square roots with a complicated procedure that worked something like a combination of long division and factoring. A slide rule made matters a lot simpler. There were long tables of square-root values. Or if you knew how to use logarithms, you could find square roots that way, too. But things are much different today. The key to finding the square root of any number--whole number, fraction, decimal--is a calculator key. Calculator square-root key

To find the square root of any number, simply key in the number (the radicand) and press the square-root key.

Examples & Exercises

 Square Roots Use a calculator to find the square roots of the given numbers. Round your answer to the nearest hundredth. Work these problems until you can do the work without making any errors.

Introducing Cube Roots

Question: What number multiplied by itself three times is equal to 8?
Since 2. x 2 x 2 = 8, so the answer is 2.

Question: What number multiplied by itself three times is equal to 64?
The answer is 4. Why? Because 4 x 4 x 4 = 64.

These are examples of finding cube roots. The cube root of 8 is 2. The cube root of 64 is 4.

 Definition The cube root of a number is a second number which, when multiplied by itself three times, equals the original number

The symbol for the cube root of a number is the radical sign with a little 3 that indicates the cube root:  The cube-root operation is the opposite of the cubing operation:

• The cube of 3 is 27 (32 = 27)
• The cube root of 27 is 3

If you can find the cube of a number, you should be able to determine the cube  root of the result.

 Cubes Cube Roots 13 = 1 1 = 1 23 = 8 8 = 2 33 = 27 27 = 3 43 = 64 64 = 4 53 = 125 125 = 5

This table  lists the Cubes for integers between 1 and 5. The Cube Roots column shows how we can use square roots to convert the squares back to their roots.

Examples

1.   Since 43 = 64, then 64 = 4
2.   Since –53 = –125, then –125 is –5
3,   Since 103 = 1000, then 1000 = 10
 Notes About Roots of Negative Numbers The cube root of zero is zero: 0 = 0 The cube root of a negative number is also a negative number. (Note that negative cube roots are okay ... negative square roots are not!)

Examples & Exercises

 Cubes and Cube Roots Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

Using a Calculator to Find Cubes and Cube Roots

You should be able to do a few, smaller-value cubes and cube roots in your head. But most of the time, however, you will need to use a calculator—especially for finding cube roots.

To find the cube root of any number, simply key in the number (the radicand) and press cube-root key. On most calculators, the cube-root function is a 2nd level function. This means you have to press the 2nd key before pressing the key for the 2nd-level key.

Examples & Exercises

 Cube Roots Use a calculator to find the cube roots of the given numbers. Round your answer to the nearest hundredth. Work with these Examples & Exercises until you can do the work without making any errors.

Rewriting Roots in Exponent Form

In previous lessons, you learned that a number multiplied by itself can be written with "square" notation.  If that number is any number n, then the square can be  shown as n2.

n n = n2

Likewise:

n n = n3 n = n1/3
Ö n = n½

Powers that are fractions represent roots of the coefficient

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