Lesson RADICALS
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Algebra: Radicals -- complicated equations involving roots
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This lesson covers an overview of RADICALS. REFERENCES <a href = "http://www.purplemath.com/" target = "_blank">http://www.purplemath.com/</a> <a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut4_radical.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut4_radical.htm</a> <a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut5_ratexp.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut5_ratexp.htm</a> <a href = "http://tutorial.math.lamar.edu/Classes/Alg/Radicals.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/Radicals.aspx</a> <a href = "http://www.themathpage.com/alg/radicals.htm" target = "_blank">http://www.themathpage.com/alg/radicals.htm</a> <a href = "http://www.themathpage.com/alg/simplify-radicals.htm" target = "_blank">http://www.themathpage.com/alg/simplify-radicals.htm</a> <a href = "http://www.themathpage.com/alg/multiply-radicals.htm" target = "_blank">http://www.themathpage.com/alg/multiply-radicals.htm</a> <a href = "http://www.themathpage.com/alg/rational-exponents.htm" target = "_blank">http://www.themathpage.com/alg/rational-exponents.htm</a> These references have lots more example and practice problems as well. TERMINOLOGY A radical is what is used to reduce something to a lower power. You would normally reference it as the square root of something or the cube root of something, or the 4th root of something or the nth root of something. Example: {{{root(3,125) = 5}}} This means the cube root of 125 = 5. The 125 is the radicand in the operation. It is shown under the root symbol. It is the number you are taking the root of. The 3 is the index of the operation. It represents the degree or power of the root. The 5 is the result of the operation usually called the root. The symbol shown is the symbol of taking the root of something. It is called the radical. DEFINITIONS Exponents and Radicals are intimately tied together. It is difficult to talk fully about one without talking about the other. That is why exponents are being discussed here as well. For more information on exponents, see the lesson on EXPONENTS. DEFINITION NUMBER 1 {{{x^n}}} This means you take x and multiply it by itself (n-1) times. n is assumed to be an integer > 0 x is the base. It is what is being raised by the exponent. n is the exponent. Example: {{{2^5}}} = 2 multiplied by itself 4 times = {{{2*2*2*2*2}}} = 32 DEFINITION NUMBER 2 {{{root(n,x)}}} This means the nth root of x. You find a number that, when raised to the power of n, equals x. n is assumed to be an integer > 0 Example: {{{root(5,32)}}} = 2 This means the 5th root of 32 = 2. The radicand is 32. The index of the radicand or the power of the root is 5. The result or root of the operation is 2. You would say that the 5th root of 32 = 2. DEFINITION NUMBER 3 {{{root(n,y) = x}}} if and only if {{{x^n = y}}} This means the nth root of y = x if and only if x to the nth power equals y. If you can lower something by the power of a root, then you have to be able to raise the result to the original number by the same power of an exponent. Example: {{{root(5,32) = 2}}} if and only if {{{2^5 = 32}}} This means that the 5th root of 32 = 2 if and only if 2 to the fifth power = 32. Another way of looking at this would be: If you can raise a number to the 5th power, you can take the 5th root of the result and get back to the same number. {{{2^5 = 32}}} and {{{root(5,32)= 2}}} which means {{{root(5,2^5) = 2}}} DEFINITION NUMBER 4 {{{x^(1/n) = root(n,x)}}} This means that raising something to the (1/n)th power is the same thing as reducing that something to the nth root. Example: {{{125^(1/3) = root(3,125)}}} = {{{5}}} This means that 125 to the (1/3)d power is the same as the cube root of 125. They both = 5. RULES RULE NUMBER 1 if n is an odd integer, then {{{(root(n,x)) = y}}} if n is an even integer, then {{{(root(n,x)) = abs(y)}}} If n is odd, then only one root for the number will exist and the sign of the radicand will be preserved. Example 1 for n is odd: y = {{{root(3,(-8))}}} = {{{-2}}} because {{{(-2)^3}}} = {{{-8}}}. The radicand was negative and the result was negative so the sign is preserved. Example 2 for n is odd: y = {{{root(3,(8))}}} = {{{2}}} because {{{(2)^3}}} = {{{8}}}. The radicand was positive and the result was positive so the sign is preserved. If n is even, then the root can be positive or negative but only the positive root is taken. The positive root is the principal or primary root. Example for n is even: y = {{{root(4,(16))}}} = {{{2}}} because {{{(2)^4}}} = {{{16}}} y = {{{root(4,(16))}}} = {{{-2}}} because {{{(-2)^4}}} = {{{16}}} while 2 roots are possible, only the positive root is taken. That's what the absolute value sign is there for: {{{abs(2)}}} is {{{2}}} {{{abs(-2)}}} is also {{{2}}} RULE NUMBER 2 If x is negative and n is even, then the nth root of x, shown as {{{root(n,x)}}}, is not real. Example: Find {{{(root(6,-64))}}} {{{(2)^6 = 64}}} and {{{(-2)^6 = 64}}} There is no real number that, when taken to the power of 6, equals -64. RULE NUMBER 3 Given the function {{{y = x^n}}} where n is even and x is positive, then x = +/- {{{root(n,x)}}} means that x can be positive or negative. This is DESPITE the second part of RULE NUMBER 1(where n is even) because the function here is asking for the value of x when y is a certain value, rather than asking for the value of y when x is a certain value. Note you are given a function and not a relation. A function requires that there is one and only one y value in the range for each x value in the domain. Example 1: You are asked to solve the function y = {{{x^2}}} You are given that y = 25. You solve for x to get: x = +/- {{{root(2,25)}}} = +/- 5. This is because {{{5^2}}} = 25, and {{{(-5)^2}}} = 25. Since you are solving for x, and not y, the equation is still a function because, while you have multiple values of x for y, you still only have one value of y for each x. This means that RULE NUMBER 1 when n is even does NOT apply to this situation. Example 2: You are asked to solve the function {{{y = root(4,x)}}}. You are given that x = 625. You solve for y to get: y = 5. Even though y could also be -5, you do not use that answer because RULE NUMBER 1 states that when you are looking for the nth root of something, and n is even, you take the absolute value of the root. The absolute value of the root is always the same as the positive root only. Since you are solving for y, and not x, the equation is still a function ONLY because RULE NUMBER 1 did not allow the negative root to be used. If the negative root was allowed as well, the equation would have been a relation instead of a function because you would have had more than one value of y for at least one value of x. Rule Number 1 is a mathematical convention meaning mathematicians got together and agreed that when they have a function that is taking the nth root of something when n is even, they would only take the positive root, and not the negative root. This convention preserves the identity of the equation as a function. At least that's my understanding. RULE NUMBER 5 {{{root(n,x)}}} * {{{root(n,y)}}} = {{{root(n,(x*y))}}} This rule states that multiplying the roots of 2 numbers together is the same as multiplying 2 numbers together and then taking the root of the result. Example: {{{root(3,125)*root(3,216)}}} = {{{root(3,(125)*(216))}}} Multiplying the roots of 2 numbers together: {{{root(3,125)}}} = {{{5}}} {{{root(3,216)}}} = {{{6}}} {{{5*6}}} = {{{30}}} Multiplying the 2 numbers together and then taking the root: {{{root(3,(125)*(216))}}} = {{{root(3,27000)}}} = {{{30}}} Either way you get the same root so the rule is confirmed. RULE NUMBER 6 {{{root(n,x)/root(n,y)}}} = {{{root(n,(x/y))}}} This rule states that dividing the roots of 2 numbers together is the same as dividing 2 numbers together and then taking the root of the result. Example: {{{root(3,216)/root(3,125)}}} = {{{root(3,(216)/(125))}}} Dividing the roots of 2 numbers together: {{{root(3,216)}}} = 6 {{{root(3,125)}}} = 5 {{{6/5}}} = 1.2 Dividing the 2 numbers together and then taking the root: {{{root(3,(216/125))}}} = {{{root(3,1.728)}}} = 1.2 Either way you get the same root so the rule is confirmed. OPERATIONS INVOLVING RADICALS RADICALS can be operated on just like variables. This means that all the rules or properties of arithmetic operations apply. These rules can be used to simplify expressions involving radicals. SIMPLIFICATION OF A RADICAL This definition was taken from Paul's Online Notes which can be found at the following website address: http://tutorial.math.lamar.edu/Classes/Alg/Radicals.aspx When a radical is simplified, the following statements are true: 1. All exponents in the radicand must be less than the index. 2. Any exponents in the radicand can have no factors in common with the index. 3. No fractions appear under a radical. 4. No radicals appear in the denominator of a fraction. EXAMPLE FOR STATEMENT NUMBER 1 All exponents in the radicand must be less than the index. {{{root(4,2^18)}}} = {{{root(4,((2^4)^4)*(2^2))}}} = {{{(2^4)*(root(4,2^2))}}} which becomes: {{{16*(root(4,2^2))}}} {{{root(4,2^18)}}} = {{{22.627417}}} {{{16*root(4,2^2)}}} = {{{22.627417}}} since you get the same solution both ways, the method for solving the equation is confirmed to be accurate. since the final form of the equation has the exponent of 2 in the radicand being less than the index of 4 in the radical (square root sign), then this expression has been reduced to a simpler form. It has not, however, been reduced to its simplest form because it still violates the conditions of statement number 2. EXAMPLE FOR STATEMENT NUMBER 2 Any exponents in the radicand can have no factors in common with the index. {{{16*root(4,2^2)}}} has factors in common because the index of 4 is divisible by the exponent of 2. {{{(root(4,2^2))}}} is equivalent to {{{(root(2,2))}}} so our equation becomes: {{{16*(root(2,2))}}} {{{16*(root(4,2^2))}}} = {{{22.627417}}} {{{16*(root(2,2))}}} = {{{22.627417}}} since you get the same solution both ways, the method for solving the equation is confirmed to be accurate. The original equation of: {{{root(4,2^18)}}} was reduced to a simpler form by making the exponent in the radicand less than the index. the equation became: {{{16*root(4,2^2)}}} which was further reduced to its simplest form by taking out any common factors between the exponent and the index. the equation became: {{{16*(root(2,2))}}} All 3 equations yielded the same answer of 22.627417 confirming that the method of converting to simpler forms was accurate. EXAMPLE FOR STATEMENT NUMBER 3 No fractions appear under a radical. {{{sqrt(x/y) = sqrt(x)/sqrt(y)}}} let x = 36 let y = 144 {{{sqrt(x/y) = sqrt(x)/sqrt(y)}}} becomes: {{{sqrt(36/144) = sqrt(36)/sqrt(144)}}} = {{{1/2}}} {{{sqrt(36/144)}}} = {{{sqrt(1/4)}}} = {{{1/2}}} {{{sqrt(36)/sqrt(144)}}} = {{{6/12}}} = {{{1/2}}} since both methods yield the same result, the method is confirmed. EXAMPLES FOR STATEMENT NUMBER 4 No radicals appear in the denominator of a fraction. Example 1 for statement number 4: {{{sqrt(5)/sqrt(7)}}} radical in the denominator has to be removed. multiply numerator and denominator by {{{sqrt(7)}}} to get: {{{(sqrt(5)*sqrt(7))/(sqrt(7)^2)}}} = {{{(sqrt(5)*sqrt(7))/7}}} since {{{sqrt(5)/sqrt(7)}}} = .845154255 and {{{(sqrt(5)*sqrt(7))/7}}} = .845154255 the method is confirmed. Example 2: {{{sqrt(5)/(sqrt(7) + sqrt(8))}}} radical in the denominator has to be removed. multiply numerator and denominator by {{{(sqrt(7) - sqrt(8))}}} {{{sqrt(5)/(sqrt(7) + sqrt(8))}}} becomes: {{{(sqrt(5)/(sqrt(7) + sqrt(8)))*((sqrt(7) - sqrt(8))/((sqrt(7) - sqrt(8))))}}} which becomes: {{{(sqrt(5)*(sqrt(7) - sqrt(8)))/(7 - 8)}}} which becomes: {{{(sqrt(5)*(sqrt(7) - sqrt(8)))/(-1)}}} the original equation of {{{sqrt(5)/(sqrt(7) + sqrt(8))}}} = .408475537 the simplified equation of {{{(sqrt(5)*(sqrt(7) - sqrt(8)))/(-1)}}} = .408475537 since they both result in the same answer, the method is confirmed.
