### Square Roots

The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number. For example, the square root of 16, denoted 161/2 or , is 4, because 42 = 4×4 = 16. The square root of 121, denoted , is 11, because 112 = 121. = 5/3, because (5/3)2 = 25/9. = 9, because 92 = 81. To take the square root of a fraction, take the square root of the numerator and the square root of the denominator. The square root of a number is always positive.

All perfect squares have square roots that are whole numbers. All fractions that have a perfect square in both numerator and denominator have square roots that are rational numbers. For example, = 9/7. All other positive numbers have squares that are non-terminating, non- repeating decimals, or irrational numbers. For example, = 1.41421356... and = 2.19503572....

### Square Roots of Negative Numbers

Since a positive number multiplied by itself (a positive number) is always positive, and a negative number multiplied by itself (a negative number) is always positive, a number squared is always positive. Therefore, we cannot take the square root of a negative number.

Taking a square root is almost the inverse operation of taking a square. Squaring a positive number and then taking the square root of the result does not change the number: = = 6. However, squaring a negative number and then taking the square root of the result is equivalent to taking the opposite of the negative number: = = 7. Thus, we conclude that squaring any number and then taking the square root of the result is equivalent to taking the absolute value of the given number. For example, = | 6| = 6, and = | - 7| = 7.

Taking the square root first and then squaring the result yields a slightly different case. When we take the square root of a positive number and then square the result, the number does not change: ()2 = 112 = 121. However, we cannot take the square root of a negative number and then square the result, for the simple reason that it is impossible to take the square root of a negative number.

### Cube Roots and Higher Order Roots

A cube root is a number that, when cubed, is equal to the given number. It is denoted with an exponent of "1/3". For example, the cube root of 27 is 271/3 = 3. The cube root of 125/343 is (125/343)1/3 = (1251/3)/(3431/3) = 25/7.
The text passage provides definitions and examples of square roots, square roots of negative numbers, and cube roots. It includes several examples but does not pose any specific questions or provide answers to questions.
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### Square Roots

The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number. For example, the square root of 16, denoted 161/2 or , is 4, because 42 = 4×4 = 16. The square root of 121, denoted , is 11, because 112 = 121. = 5/3, because (5/3)2 = 25/9. = 9, because 92 = 81. To take the square root of a fraction, take the square root of the numerator and the square root of the denominator. The square root of a number is always positive.

All perfect squares have square roots that are whole numbers. All fractions that have a perfect square in both numerator and denominator have square roots that are rational numbers. For example, = 9/7. All other positive numbers have squares that are non-terminating, non- repeating decimals, or irrational numbers. For example, = 1.41421356... and = 2.19503572....

### Square Roots of Negative Numbers

Since a positive number multiplied by itself (a positive number) is always positive, and a negative number multiplied by itself (a negative number) is always positive, a number squared is always positive. Therefore, we cannot take the square root of a negative number.

Taking a square root is almost the inverse operation of taking a square. Squaring a positive number and then taking the square root of the result does not change the number: = = 6. However, squaring a negative number and then taking the square root of the result is equivalent to taking the opposite of the negative number: = = 7. Thus, we conclude that squaring any number and then taking the square root of the result is equivalent to taking the absolute value of the given number. For example, = | 6| = 6, and = | - 7| = 7.

Taking the square root first and then squaring the result yields a slightly different case. When we take the square root of a positive number and then square the result, the number does not change: ()2 = 112 = 121. However, we cannot take the square root of a negative number and then square the result, for the simple reason that it is impossible to take the square root of a negative number.

### Cube Roots and Higher Order Roots

A cube root is a number that, when cubed, is equal to the given number. It is denoted with an exponent of "1/3". For example, the cube root of 27 is 271/3 = 3. The cube root of 125/343 is (125/343)1/3 = (1251/3)/(3431/3) = 25/7.
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The text passage provides definitions and examples of square roots, square roots of negative numbers, and cube roots. It includes several examples but does not pose any specific questions or provide answers to questions.

TYPE: NO_QUESTION