The notation above read as the nth root of the mth power of a, is a


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1 Let s Reduce Radicals to Bare Bones! (Simplifying Radical Expressions) By Ana Marie R. Nobleza The notation above read as the nth root of the mth power of a, is a radical expression or simply radical. This notation was introduced to let expressions with noninteger exponents a m n ( ) make sense! In your younger years, you must have known that exponents tell us how many times the base is to be taken as a factor. If that idea is to be applied to terms with exponents that are fractions like 1, would it seem sensible? Of course not, because how can you show (the base) taken 1 times as a factor? This calls for radicals because the idea of exponents mentioned above is only applicable to positive integral exponents. Radical expressions occur in quite a number of scientific and mathematical formulas. One simple example of these formulas is s = A, which gives us the length of the sides of a square given its area. Sometimes or even often, using these formulas leads us to irrational radicals, whose values could only be approximated by using a calculator or computer. What if we were required to express the result of our computation in exact simplified form? For this situation, we would need to learn how to simplify radical expressions. When can we say that a radical expression is already simplified? FOR SINGLE TERM EXPRESSIONS We can say that a single term radical expression is already in its simplest form if all of the following conditions are satisfied: The radicand contains no factor from which the indicated root can be taken. The radicand is not a fraction. The term contains no radical in the denominator. TATSULOK First Year Vol. 1 No. 1a 6b 1
2 The order of the radical is lowest, meaning, its index divided by the power of the radicand gives a proper fraction in lowest term. Before we illustrate the conditions above, let us study the following rules for radicals, which will help us in simplifying them. PRODUCT RULE FOR RADICALS For all real numbers a and b for which the radicals are defined (no negative radicand for even indices), and all positive integers n, n n n ab = a b In layman s term, this means that the nth root of a product is equal to the product of the nth roots of its factors. QUOTIENT RULE FOR RADICALS For all real numbers a and b with b 0, and for which the radicals are defined, and all positive integers n, n a a n = n b b In layman s term, this means that the nth root of a quotient is equal to the quotient of the nth roots of its dividend and divisor. To illustrate the conditions presented above, let us have the following examples. Example 1: Simplify x y. Step 1: Prime factorize or simply factor the radicand to check for perfect square factors (we are given a square root term so we need to scout for perfect squares in the factors): x y or 16 x y y Step : Apply the product rule and extract the square roots of the perfect square factors and let the nonperfect square factors remain under the radical sign. 16 x y y = xy y Example : Simplify 00x Step 1: Factor the radicand and extract the perfect fourth power from the factors, leaving the nonperfect factors under the radical sign. x = x TATSULOK First Year Vol. 1 No. 1a 6b
3 Step : Since the remaining radical term is not yet in lowest term, divide the index and the power of the radicand by their greatest common factor, in this case,. Therefore, the final simplified expression must be: x = x = x Example : Simplify Step 1: For radicals whose radicands are fractions always make the denominator a perfect power (in this case perfect square) by multiplying the numerator and denominator by an appropriate number. Here we have: 1 = or Step : Apply the quotient rule and take out the roots that can be extracted. Thus, we have: 1 = or 1 1 = or We also call this process shown in example rationalizing the denominator. There is a separate article for this topic. MULTIPLE TERM EXPRESSIONS When the radical expressions that need to be simplified contain two or more terms, we first apply the processes for simplifying single term expressions to simplify each of the terms (if needed) and then combine similar radicals. Radicals are considered similar if they have the same indices and radicand. ADDITION AND SUBTRACTION OF RADICAL EXPRESSIONS Combining similar radicals may be compared to combining similar algebraic terms where we add or subtract numerical coefficients and then affix to the sum or difference the common literal coefficients. We combine radicals as if we are combining similar terms of polynomials or any algebraic expressions. To better understand this process, let us study the following examples: Example : Simplify 0 00 Step 1: Simplify each term of the expression by applying the appropriate process for simplifying single term expressions (discussed above). For this example, we have: = = + 10 or TATSULOK First Year Vol. 1 No. 1a 6b
4 Step : Combine similar terms. In this case, we can see that all the terms are similar, thus we have: = 16 Example : Simplify: We follow the same steps as in example to arrive at the following: = = = + 10 MULTIPLICATION OF RADICAL EXPRESSIONS Simplifying radical expressions may also entail multiplication or division of terms. We multiply radicals in the same manner as when we multiply algebraic expressions where we apply the distributive property of multiplication, but division of radicals is different. This is discussed in the article for rationalizing denominators. To illustrate multiplication of radical expressions, let us have the following examples: ( ) ( ) Example 6: Simplify 8 ( )( ) ( 8 ) ( ) = 16 or 1 Example 7: Simplify + Step 1: We use the FOIL or distributive method here to have: ( )( ) = + ( ) + 1 Step : Simplify and combine similar terms: ( + )( ) = 1 + ( ) = ( 1 ) + = 1 + TATSULOK First Year Vol. 1 No. 1a 6b
5 WORKSHEET In Math, we know that a radical has something to do with roots. But did you know that in the English language, the term radical has multiple meanings? Discover one of its meanings by simplifying the radical expressions in each box below. When you are through, write the letter of the expression that corresponds to each simple term under each line and then you ll have it! x 1 x 6 x 1 (This meaning also refers to a person who takes militant action in the service of a party or doctrine.) A C S 16x 8 7 E 100 D + T ( + )( ) I F V 6 x O ( ) G 18x x W ( 1 ) U 00 H x 16x X ( + ) ANSWER: ACTIVIST TATSULOK First Year Vol. 1 No. 1a 6b
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