Square Roots and Cube Roots

Note: Square roots and cube roots and operations with them are often included in algebra books.

To find the square root of a number, you want to find some number that when multiplied by itself gives you the original number. In other words, to find the square root of 25, you want to find the number that when multiplied by itself gives you 25. The square root of 25, then, is 5. The symbol for the square root is equation. Following is a partial list of perfect (whole number) square roots.

equation

Note: If no sign (or a positive sign) is placed in front of the square root, the positive answer is required. No sign means that a positive is understood. Only if a negative sign is in front of the square root is the negative answer required. This notation is used in many texts, as well as this book. Therefore, equation

To find the cube root of a number, you want to find some number that when multiplied by itself twice gives you the original number. In other words, to find the cube root of 8, you want to find the number that when multiplied by itself twice gives you 8. The cube root of 8, then, is 2 because 2 × 2 × 2 = 8. Notice that the symbol for cube root is the radical sign with a small three (called the index) above and to the left, equation. Other roots are similarly defined and identified by the index given. (In square root, an index of 2 is understood and is usually not written.) Following is a partial list of perfect (whole number) cube roots.

equation

To find the square root of a number that is not a perfect square, it is necessary to find an approximate answer by using the procedure given in Example .

Approximate equation.

equation is between equation and equation

equation

equation and equation

Therefore, equation

Since 42 is almost halfway between 36 and 49, equation is almost halfway between equation and equation. So equation is approximately 6.5. To check, multiply the following:

6.5 × 6.5 = 42.25 or about 42.

Approximate equation.

equation

Since equation is slightly closer to equation than it is to equationequation

Check the answer.

equation

Approximate equation.

First, perform the operation under the radical.

equation

Since equation is slightly closer to equation than it is to equation.

equation

Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind, because they are commonly used.

equation

Sometimes you will have to simplify square roots, or write them in simplest form. In fractions, equation can be simplified to equation. In square roots, equation can be simplified to equation.

There are two main methods to simplify a square root.

Method 1:

Factor the number under the equation into two factors, one of which is the largest possible perfect square. (Perfect squares are 1, 4, 9, 16, 25, 36, 49, and so on)

Method 2:

Completely factor the number under the equation into prime factors and then simplify by bringing out any factors that came in pairs.

Simplify equation.

Method 1.

equation

Take the square root of the perfect square number

equation

Finally, write it as a single expression.

equation

Method 2.

equation

Rewrite with pairs under the radical

equation

In Example , the largest perfect square is easy to see, and Method 1 probably is a faster method.

Simplify equation.

Method 1.

equation

Method 2.

equation

In Example , it is not so obvious that the largest perfect square is 144, so method 2 is probably the faster method.

Simplify equation.

Method 1.

equation

Method 2.

equation

Remember: Most square roots cannot be simplified since they are already in simplest form, such as equationequationequation.