Overview
Radical expressions involving square roots have many applications in algebra. If an equation calls for a variable to be squared, its solution will be a square root. The principal square root is a positive number.
What Are Perfect Squares?
Perfect squares have whole numbers as square roots. For example, √9 equals 3, and √16 equals 4. That is because 32=9 and 42=16. In addition, both 9 and 16 have negative square roots, -3 and -4, but they are not usually considered. Sometimes perfect squares can be found by prime factorizations. For example, 576 can be factored as 26∙∙ 32. The square root of 26 is 23, because 23∙23 is 26, and the square root of 32 is 3. The square root of 576 is then 8 ∙ 3, or 24.
What Are Irrational Numbers?
Most numbers, however, are not perfect squares. For example, what is √10? Prime factorization will not help in this case, because neither √5 or√2 are perfect squares. The √10 is approximately 3.16227766, and the decimal neither repeats or terminates. Decimals that neither repeat or terminate are irrational numbers, because there is no fraction that exactly expresses them.
When Are Radical Expressions in Simplest Form?
Radical expressions are in simplest form if all perfect square factors other than 1 have been simplified. According to the Product Property of Square Roots, √ab = √a∙√b for any numbers a and b when both a and b are greater than zero. Therefore, √20 can be simplified to √4∙√5, or 2√5. Similarly, √20∙√10 can be simplified to √10∙10∙2 or 10√2.
How to Add or Subtract Radical Expressions?
In order to add or subtract radical expressions, the radicands (the numbers under the radical sign) must be alike and in simplest form. For example, 9√7-4√2 + 3√2 + 5√7 can be simplified using the commutative property to 14√7-√2. Suppose the expression was 3√20 + 4√5 + 5√3. It could be simplified to 3( 2√5) + 4√5 + 5√3 or 6√5 + 4√5 + 5√3 or 10√5 + 5√3.
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