Section P.3 Radicals and Rational Exponents

Simplifying Expressions of the Form √a ²

( 2. Simplify expressions of the form squareroot(a²)) You may think that sq.root(a²) = a. However, this is not necessarily true. Consider the following examples: sq.root(4²) = sq.root(16) = 4. sq.root(-4) = sq.root(16) = 4 (the result is not -4, but rather the absolute value of -4, or 4. Here is a rule for simplifying expressions of the form sq.root(a²): Simplifying sq.root(a²) ------------------------ For any real number "a", sq.root(a²) = | a |. In words, the principal square root of a² is the absolute value of "a". For example, sq.root(6²) = | 6 | = 6. sq.root((-6)²) = | -6 | = 6.

Square Roots.

(1 Evaluate square roots.) From our earlier work with exponents, we are aware that the square of both 5 and (-5) is 25. The reverse operation of squaring a number is finding the square root of the number. For example: • One square root of 25 is 5 because 5² = 25. • Another square root of 25 is -5 because (-5)² = 25. In general, if b² = a, then b is a square root of a. The Radical Sign is used to denote the nonnegative or principal square root of a number. For example: • square root(25) = 5 because 5² = 25 and 5 is positive. • square root(100) = 10 because 10² = 100 and 10 is a positive number. The symbol that we use to denote the principal square root is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as a radical expression. ​

The Product Rule for Square Roots.

(3 Use the product rule to simplify square roots.) A rule for multiplying square roots can be generalized by comparing: sq.root(25) • sq.root(4) and sq.root(25 • 4). Notice that sq.root(25) = 5; sq.root(4) = 2, thus 5 • 2 = 10. and sq.root(25 • 4) = sq.root(100) = 10. Because we obtain 10 in both situations, the original radical expressions must be equal. That is, sq.root(25) • sq.root(4) = sq.root(25 • 4). This result is a special case of the product rule for square roots that can be generalized as follows: The Product Rule for Square Roots: --------------------------------------- If a and b represent nonnegative real numbers, then sq.root(ab) = sq.root(a) • sq.root(b) [ The square root of a product is the product of the square roots] and sq.root(a) • sq.root(b) = sq.root(ab) [The product of two roots is the square of the product of the roots.] A square root is simplified when its radicand has no factors other than 1 that are perfect squares. For example, sq.root(500) is not simplified because it can be expressed as sq.root(100 • 5) and 100 is a perfect square. Example 2 shows how the product rule is used to remove from the square root any perfect squares that occur as factors. ​

The Quotient Rule for Square Roots.

(4 Use the quotient rule to simplify square roots.) Another property for square roots involves division. If a and b represent nonnegative real numbers and b ≠ 0, then sq.root(a/b) = sq.root(a) / sq.root(b) [The square root of a quotient is the quotient of the square roots.] and sq.root(a) / sq.root(b) = sq.root(a/b) [The quotient of two square roots is the square root of the quotient of the radicands.]

Adding and Subtracting Square Roots.

(5 Add and subtract square roots.) Two or more square roots can be combined using the distributive property provided that they have the same radicand. Such radicals are called like radicals. For example: 7 square roots of 11 plus 6 square roots of 11 result in 13 square roots of 11.

Rationalizing Denominators.

(6 Rationalize denominators.) The calculator shows approximate values for 1/sq root(3) and sq.root(3)/3. The two approximations are the same. This is not a coincidence. 1/sq.root(3) = 1/sq.root(3) • sq.root(3)/sq.root(3) = sq.root(3)/sq.root(9) = sq.root(3)/3. Any number divided by itself is 1. Multiplication by 1 does not change the value of 1/sq.root(3). This process involves rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals. The process is called rationalizing the denominator. If the denominator consists of the square root of a natural number that is not a perfect square, multiply the numerator and the denominator by the smallest number that produces the square root of a perfect square in the denominator. example: a.) 15/sq.root(6) If we multiply the numerator and the denominator of 15/sq.root(6) by sq.root(6), the denominator becomes sq.root(6) • sq.root(6) = sq.root(36) = 6. Therefore, we multiply by 1, choosing sq.root(6)/sq.root(6) for 1. 15/sq.root(6) = 15/sq.root(6) • sq.root(6)/sq.root(6) = 15.sq.root(6)/sq.root(36) = 15.sq.root(6)/6 = 5.sq.root(6)/2.

Other Kinds of Roots.

(7 Evaluate and perform operations with higher roots.) We define the principal nth root of a real number "a", symbolized by n.sq.root(a), as follows: n.sq.root(a) = b means that b^n = a. If n, the index, is even, then "a" is nonnegative (a >= 0) and "b" is also nonnegative (b >=0). If "n" is odd, "a" and "b" can be any real numbers. For example, 3.sq.root(64) = 4 because 4³ = 64 and 5.sq.root(-32) = -2 because (-2)⁵ = -32. The same vocabulary that we learned for square roots applies to nth roots. The symbol n.sq.root is called a radical and the expression under the radical is called the radicand. A number that is the nth power of a rational number is called a perfect nth power. For example, 8 is a perfect third power, or perfect cube, because 8 = 2³. Thus, 3.sq.root(8) = 3.sq.root(2³) = 2. In general, one of the following rules can be used to find the nth root of a perfect nth power.

Rational Exponents

(8 Understand and use rational exponents.) We define rational exponents so that their properties are the same as the properties for integer exponents. For example, we know that exponents are multiplied when an exponential expression is raised to a power. For this to be true, (7 • 1/2) = 7 • 1/2 • 2 = 7¹ = 7. We also know that, (sq.root(7)² = sq.root(7) • sq.root(7) = sq.root(49) = 7. Can you see that the square of both 7 • 1/2 and sq.root(7) is 7? It is reasonable to conclude that 7•1/2 means sq.root(7). We can generalize the fact that 7•1/2 means sq.root(7) with the following definition: The Definition of a•1/n: If n.sq.root(a) represents a real number, where n >= 2 is an integer, then a•1/2 = n.sq.root(a). [The denominator of the rational exponent is the radical's index. Furthermore, a • -1/n = 1/a•1/n = 1/n.sq.root(a), a ≠ 0.

Multiplying Conjugates

(sq.root(a) + sq.root(b)) (sq.root(a) - sq.root(b)) = (sq.root(a)² - (sq.root(b))² = a - b. How can we Rationalize a denominator if the denominator contains two terms with one or more square roots? Multiply the numerator and the denominator by the conjugate of the denominator. Here are three examples of such expressions: 7/ 5+sq.root(3) • The conjugate of the denominator is 5 - sq.root(3). 8/3.sq.root(2-4) • The conjugate of the denominator is 3.sq.root(2+4) h/sq.root(x + h - sq.root(x) • The conjugate of the denominator is sq.root(x + h - sq.root(x). The product of the denominator and its conjugate is found using the formula: (sq.root(a) + sq.root(b)) (sq.root(a) - sq.root(b)) = (sq.root(a))² - (sq.root(b))² = a - b. The simplified product will not contain a radical.

The Products and Quotient Rules for nth Roots.

For all real numbers "a" and "b", where the indicated roots represent real numbers, n.sq.root(ab) = n.sq.root(a) • n.sq.root(b) [The nth root of a product is the product of the nth roots] —and— n.sq.root(a) • n.sq.root(b) = n.sq.root(ab) [The product of two nth roots is the nth root of the product of the radicands.] n.sq.root(a/b) = n.sq.root(a) / n.sq.root(b), b ≠ 0. [The nth root of a quotient is the quotient of the nth roots.] and n.sq.root(a) / n.sq.root(b) = n.sq.root(a/b) , b ≠ 0. [ The quotient of two nth roots is the nth root of the quotient of the radicands.] We have seen that adding and subtracting square roots often involves simplifying terms. The same idea applies to adding and subtracting higher roots. example: pg. 44

What exactly does rationalizing a denominator do to an irrational number in the denominator?

Rationalizing a numerical denominator makes that denominator a rational number.

Definition of the Principal Square Root.

If "a" is a nonnegative real number, the nonnegative number "b" such that b² = a, denoted by b = squareroot(a), is the principal square root of a. The symbol, negative squareroot, is used to denote the negative square root of a number. For example: • negative.squareroot(25) = -5 because (-5)² = 25 and -5 is negative. • negative.squareroot(100) = -10 because (-10)² = 100 and -10 is negative.

Finding the nth roots of Perfect nth Powers

If n is odd, n.sq.root(a^n) = a. If n is even, n.sq.root(a^n) = | a |. For example: 3.sq.root((-2)³) = -2 and 4.sq.root((-2)⁴) = | -2 | = 2. Absolute value is not needed with odd roots, but is necessary with even roots.

Is the squareroot(a+b) = squareroot(a) + squareroot(b)?

No. In Example 1, parts (d) and (e), observe that squareroot(9 + 16) is not equal to squareroot(9) + squareroot(16). In general: squareroot(a + b) ≠ squareroot(a) + squareroo(b) and squareroot(a - b) ≠ squareroot (a) - squareroot(b).

Should I know the higher roots of certain numbers?

Some higher roots occur so frequently that you might want to memorize them.

Example 2 Using the Product Rule to Simplify Square Roots.

Simplify sq.root(500): (1) Factor 500. 100 is the greatest perfect square factor = sq.root(100•5) (2) Using the product rule: sq.root(ab) = sq.root(a) • sq.root(b). Thus = sq.root(100) • sq.root(5). (3) Write sq.root(100) as 10. We read 10.sq.root(5) as "ten times the square root of 5." = 10.sq.root(5) Simplify sq.root(6x) • sq.root(3x) We can simplify sq.root(6x) • sq.root(3x) using the product rule only if 6x and 3x represent nonnegative real numbers. Thus, x >= 0. (1) Use the product rule: sq.root(a) • sq.root(b) = sq.root(ab). Thus = sq.root(6x) • sq.root(3x) = sq.root(6x • 3x) (2) Multiply in the radicand = sq.root(18x²). (3) Factor 18. 9 is the greatest perfect square factor. = sq.root(9x²•2) (4) Use the product rule: sq.root(ab) = sq.root (a) • sq.root(b) = sq.root(9x²) • sq.root(2). (5) Use the product rule to write sq.root(9x²) as the product of two squares. = sq.root(9) • sq.root(x²) • sq.root(2). (6) sq.root(x²) = | x | = x because x >= 0. Thus = 3x.sq.root(2).

The Product and Quotient Rules for Other Roots.

The product and quotient rules apply to cube roots, fourth roots, and all higher roots.

Should like radicals remind me of like terms?

Yes. Adding or subtracting like radicals is similar to adding or subtracting like terms: 7x + 6x = 13x and 7 sq.roots of 11 plus 6 sq.roots of 11 result in 13 sq.roots of 11.

When simplifying square roots, what happens if I use a perfect square factor that isn't the greatest perfect square factor possible?

You'll need to simplify even further. For example, consider the following factorization: sq.root(500) = sq.root(25•20) = sq.root(25) • sq.root(20) = 5.sq.root(20). Because 20 contains a perfect square factor, 4, the simplification is not complete. 5.sq.root(20) = 5.sq.root(4 • 5) = sq.root(4) • sq.root(5) = 5•2.sq.root(5) = 10.sq.root(5) Although the result checks with our simplification using sq.root(500) = sq.root(100•5), more work is required when the greatest perfect square factor is not used.