Section 5.4 The Irrational Numbers Math in Our World

Learning Objectives  Define irrational numbers.  Simplify radicals.  Multiply and divide square roots.  Add and subtract square roots.  Rationalize denominators.

Irrational Numbers Here’s an example: We can see the pattern—a zero followed by an odd number, with the odds increasing each time. But this pattern continues forever, so the decimal is not terminating, and there’s no definite string that repeats, so it’s not repeating either. A number is irrational if it can be written as a decimal that neither terminates nor repeats.

Square Roots For example, the number √2 = is irrational. The symbol √2 is read as “the square root of two,” and the symbol around 2 is called a radical sign. The square root of a number a, symbolized, is the nonnegative number you have to multiply by itself (or square) to get a.

Square Roots Not every square root is irrational. Since 2 2 = 4, √4 = 2, so √4 is an integer, and consequently a rational number. The same is true for √9, √16, √25, and so on. In each case, the number under the radical is the square of an integer. We will call such a number a perfect square, and observe that the square root of a perfect square is always an integer. But for any other number, the square root happens to be irrational.

Simplifying Radicals The Product Rule For Square Roots For any two positive numbers a and b, Notes: The product rule is also true if a and/or b are equal to zero, but it’s not particularly useful in that case. This is called the product rule for a good reason—a similar formula is not true for all operations. Most notably,

EXAMPLE 1 Simplify Each Radical Simplify each radical. (a) (b) (c)

EXAMPLE 1 Simplify Each Radical SOLUTION (a) Notice that 40 can be written as 4 10, and that 4 is a perfect square. Also, 4 is the largest factor of 40 that is a perfect square. Using the product rule, we can write (b) The largest perfect square factor of 200 is 100. (c) The prime factorization of 26 is It has no perfect square factors, so √26 is already simplified.

EXAMPLE 2 Multiplying Square Roots Find each product. (a) (b) (c)

EXAMPLE 2 Multiplying Square Roots SOLUTION (a) Using the product rule (reading it from right to left), we can write the product as a single square root by multiplying the numbers under the radical. Now we can simplify like we did in Example 1. (b) Again, we can multiply the two numbers under the radical. It is equal to 10 because 10 2 = 100. This shows that the product of two irrational numbers is not always irrational.

EXAMPLE 2 Multiplying Square Roots SOLUTION (c) The product rule doesn’t specifically tell us how to multiply three radicals, but we can use it in stages (multiply the first two, then multiply the result by the third) to show that we can multiply the three numbers inside the radical. Now we can simplify. In this case, it would have been simpler to notice that the product of the second and third factors, √3 √3, is 3, in which case we would have obtained the same answer, but much more quickly.

Simplifying Radicals The Quotient Rule For Square Roots For any two positive numbers a and b, Note: The quotient rule is also true if a = 0, but not if b = 0, as this would make the denominator zero.

EXAMPLE 3 Using the Quotient Rule Find each quotient. (a)(b) SOLUTION In each case, we’ll use the quotient rule to write the quotient as a single square root, dividing the numbers underneath the radical, then simplify. (a)(b)

Like Radicals In the expression below, the number under the radical is known as the radicand, and the number in front of the radical is called the coefficient. When the radicands of two or more different square roots are the same, we call them like radicals. Coefficient Radicand

Adding & Subtracting Radicals Addition & Subtraction of Like Radicals To add or subtract like radicals, add or subtract their coefficients and keep the radical the same. In symbols,

EXAMPLE 4 Adding Square Roots Find the sum: SOLUTION Since all the radicals are like radicals, the sum can be found by adding the coefficients of the radicals.

EXAMPLE 5 Subtracting Square Roots Find the difference: SOLUTION

EXAMPLE 6 Adding and Subtracting Square Roots Perform the indicated operations: SOLUTION Simplify each radical first, and then add or subtract like radicals as shown. Now we see that we have like radicals.

Rationalizing the Denominator Another method used to simplify radical expressions is called rationalizing the denominator. When a radical expression contains a square root sign in the denominator of a fraction, it can be simplified by multiplying the numerator and denominator by a radical expression that will make the radicand in the denominator a perfect square.

EXAMPLE 7 Rationalizing Denominators Simplify each radical expression. (a)(b)

EXAMPLE 7 Adding & Subtracting Fractions with a Common Denominator SOLUTION (a)If we multiply the numerator and denominator by √3, the denominator will become √9, which is of course 3. (b) We could mimic what happened in part (a) and multiply the numerator and denominator by √18, but it will be easier to simplify if we multiply instead by √2. This will make the denominator √36, which is 6.

EXAMPLE 8 Simplifying the Square Root of a Fraction Simplify: SOLUTION Apply the quotient rule to split into two separate roots: Now multiply the numerator and denominator by √6 to rationalize the denominator: