The root form is an algebraic statement that has the sign of the square root (or cube root or higher). This form can often represent two numbers that have the same value even though they may appear different at first glance (for example, 1/(sqrt(2) - 1) = sqrt(2)+1). Therefore, we need a "standard formula" for this kind of form. If there are two statements, both in the standard formula, that appear different, they are not the same. Mathematicians agree that the standard formulation of the quadratic form satisfies the following requirements:
- Avoid using fractions
- Do not use fractional powers
- Avoid using the root form in the denominator
- Does not contain the multiplication of two root forms
- Numbers under the root cannot be rooted anymore
One practical use of this is in multiple choice exams. When you find an answer, but your answer is not the same as the available choices, try to simplify it into a standard formula. Since question makers usually write answers in standard formulas, do the same with your answers to match theirs. In essay questions, commands such as "simplify your answer" or "simplify all root forms" mean that students have to carry out the following steps until they meet the standard formula as above. This step can also be used to solve equations, although some types of equations are easier to solve in non-standard formulas.
Step
Step 1. If necessary, review the rules for operating roots and exponents (both are equal - roots are powers of fractions) as we need them in this process
Also review the rules for simplifying polynomials and rational forms as we will need to simplify them.
Method 1 of 6: Perfect Squares
Step 1. Simplify all roots containing perfect squares
A perfect square is the product of a number by itself, for example 81, which is a product of 9 x 9. To simplify a perfect square, just remove the square root and write down the square root of the number.
- For example, 121 is a perfect square because 11 x 11 equals 121. So, you can simplify the root(121) to 11, by removing the root sign.
- To make this step easier, you'll need to remember the first twelve perfect squares: 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, 6 x 6 = 36, 7 x 7 = 49, 8 x 8 = 64, 9 x 9 = 81, 10 x 10 = 100, 11 x 11 = 121, 12 x 12 = 144
Step 2. Simplify all roots containing perfect cubes
A perfect cube is the result of multiplying a number by itself twice, for example 27, which is the product of 3 x 3 x 3. To simplify the root form of a perfect cube, just remove the square root and write down the square root of the number.
For example, 343 is a perfect cube because it is the product of 7 x 7 x 7. So the cube root of 343 is 7
Method 2 of 6: Converting Fractions to Roots
Or changing the other way around (it helps sometimes), but don't mix them up in the same statement as root(5) + 5^(3/2). We'll assume that you want to use the root form and we'll use the symbols root(n) for the square root and sqrt^3(n) for the cube root.
Step 1. Take a fraction to the power of a fraction and convert it to the root form, for example x^(a/b) = root to the b power of x^a
If the square root is in fraction form, convert it to regular form. For example, square root (2/3) of 4 = root(4)^3 = 2^3 = 8
Step 2. Convert negative exponents to fractions, for example x^-y = 1/x^y
This formula only applies to constant and rational exponents. If you're dealing with a form like 2^x, don't change it, even if the problem indicates that x can be a fraction or a negative number
Step 3. Merge same tribe and simplify the resulting rational form.
Method 3 of 6: Eliminating Fractions in Roots
The standard formula requires that the root be an integer.
Step 1. Look at the number under the square root if it still contains a fraction
If still,…
Step 2. Replace it with a fraction consisting of two roots using the identity sqrt(a/b) = sqrt(a)/sqrt(b)
Do not use this identity if the denominator is negative, or if it is a variable that might be negative. In this case, simplify the fraction first
Step 3. Simplify each perfect square of the result
That is, convert sqrt(5/4) to sqrt(5)/sqrt(4), then simplify to sqrt(5)/2.
Step 4. Use other simplification methods such as simplifying complex fractions, combining equal terms, etc
Method 4 of 6: Combining Multiplication Roots
Step 1. If you are multiplying one root form by another, combine the two in one square root using the formula:
sqrt(a)*sqrt(b) = sqrt(ab). For example, change sqrt(2)*sqrt(6) to sqrt(12).
- The identity above, sqrt(a)*sqrt(b) = sqrt(ab), is valid if the number under the sign of the sqrt is not negative. Do not use the formula when a and b are negative because you will make the mistake of making sqrt(-1)*sqrt(-1) = sqrt(1). The statement on the left is equal to -1 (or undefined if you don't use complex numbers) while the statement on the right is +1. If a and/or b are negative, first "change" the sign like sqrt(-5) = i*sqrt(5). If the form under the root sign is a variable whose sign is unknown from the context or can be positive or negative, leave it as is for the time being. You can use the more general identity, sqrt(a)*sqrt(b) = sqrt(sgn(a))*sqrt(sgn(b))*sqrt(|ab|) which applies to all real numbers a and b, but usually this formula doesn't help much because it adds complexity to using the sgn (signum) function.
- This identity is valid only if the forms of the roots have the same exponent. You can multiply different square roots such as sqrt(5)*sqrt^3(7) by converting them to the same square root. To do this, temporarily convert the cube root to a fraction: sqrt(5)*sqrt^3(7) = 5^(1/2) * 7^(1/3) = 5^(3/6) * 7 ^(2/6) = 125^(1/6) * 49^(1/6). Then use the multiplication rule to multiply the two to the square root of 6125.
Method 5 of 6: Removing the Square Factor from the Root
Step 1. Factoring imperfect roots into prime factors
A factor is a number that when multiplied by another number forms a number -- for example, 5 and 4 are two factors of 20. To break down imperfect roots, write down all the factors of the number (or as many as possible, if the number is too large) until you have find a perfect square.
For example, try to find all the factors of 45: 1, 3, 5, 9, 15, and 45. 9 is a factor of 45 and is also a perfect square (9=3^2). 9 x 5 = 45
Step 2. Remove all multipliers that are perfect squares from within the square root
9 is a perfect square because it is the product of 3 x 3. Take 9 out of the square root and replace it with 3 in front of the square root, leaving 5 inside the square root. If you "put" 3 back into the square root, multiply by itself to make 9, and if you multiply by 5 it returns 45. 3 roots of 5 is a simple way of expressing the root of 45.
That is, sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5)
Step 3. Find the perfect square in the variable
The square root of a squared is |a|. You can simplify this to just "a" if the known variable is positive. The square root of a to the power of 3 when broken down to the square root of a squared times a -- remember that the exponents add up when we multiply two numbers to the power of a, so a squared times a equals a to the third power.
Therefore, a perfect square in the form a cubed is a squared
Step 4. Remove the variable containing the perfect square from the square root
Now, take a squared from the square root and change it to |a|. The simple form of the root of a to the power of 3 is |a| root a.
Step 5. Combine the equal terms and simplify all the roots of the calculation results
Method 6 of 6: Rationalize the Denominator
Step 1. The standard formula requires that the denominator be an integer (or a polynomial if it contains a variable) as much as possible
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If the denominator consists of one term under the root sign, such as […]/root(5), then multiply both the numerator and denominator by that root to get […]*sqrt(5)/sqrt(5)*sqrt(5) = […]*root(5)/5.
For cube roots or higher, multiply by the appropriate root so that the denominator is rational. If the denominator is root^3(5), multiply the numerator and denominator by sqrt^3(5)^2
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If the denominator consists of adding or subtracting two square roots such as sqrt(2) + sqrt(6), multiply the quantifier and denominator by their conjugate, which is the same form but with the opposite sign. Then […]/(root(2) + root(6)) = […](root(2)-root(6))/(root(2) + root(6))(root(2)-root (6)). Then use the identity formula for the difference of two squares [(a+b)(ab) = a^2-b^2] to rationalize the denominator, to simplify (sqrt(2) + sqrt(6))(sqrt(2)-sqrt(6)) = sqrt(2)^2 - sqrt(6)^2 = 2-6 = -4.
- This also applies to denominators like 5 + sqrt(3) because all integers are roots of other integers. [1/(5 + sqrt(3)) = (5-sqrt(3))/(5 + sqrt(3))(5-sqrt(3)) = (5-sqrt(3))/(5^ 2-sqrt(3)^2) = (5-sqrt(3))/(25-3) = (5-sqrt(3))/22]
- This method also applies to the addition of roots such as sqrt(5)-sqrt(6)+sqrt(7). If you group them into (sqrt(5)-sqrt(6))+sqrt(7) and multiply by (sqrt(5)-sqrt(6))-sqrt(7), the answer is not in rational form, but still in a+b*root(30) where a and b are already rational numbers. Then repeat the process with the conjugates a+b*sqrt(30) and (a+b*sqrt(30))(a-b*sqrt(30)) will be rational. In essence, if you can use this trick to remove one root sign in the denominator, you can repeat it many times to remove all the roots.
- This method can also be used for denominators that contain a higher power root such as the fourth root of 3 or the seventh root of 9. Multiply the numerator and denominator by the conjugate of the denominator. Unfortunately, we can't directly get the conjugate of the denominator and it's difficult to do that. We can find the answer in an algebra book on number theory, but I won't go into that.
Step 2. Now the denominator is in rational form, but the numerator looks a mess
Now all you have to do is multiply it by the conjugate of the denominator. Go ahead and multiply as we would multiply polynomials. Check to see if any terms can be omitted, simplified, or combined, if possible.
Step 3. If the denominator is a negative integer, multiply both the numerator and denominator by -1 to make it positive
Tips
- You can search online for sites that can help simplify root forms. Just type the equation with the root sign, and after pressing Enter, the answer will appear.
- For simpler questions, you may not use all the steps in this article. For more complex questions, you may need to use several steps more than once. Use the "simple" steps a few times, and check to see if your answer fits the standard formulation criteria we discussed earlier. If your answer is in the standard formula, you are done; but if not, you can check one of the steps above to help you get it done.
- Most references to the "recommended standard formula" for roots also apply to complex numbers (i = root(-1)). Even if a statement contains an "i" instead of a root form, avoid denominators that still contain an i as much as possible.
- Some of the instructions in this article assume all roots are in square form. The same general principles apply to the roots of higher powers, although some parts (especially rationalizing the denominator) can be quite difficult to work with. Decide for yourself what shape you want, such as sqr^3(4) or sqr^3(2)^2. (I don't remember what kind of form is usually suggested in textbooks).
- Some of the instructions in this article use the word "standard formula" to describe "regular form". The difference is that the standard formula only accepts the form 1+sqrt(2) or sqrt(2)+1 and considers the other forms as non-standard; Plain form assumes that you, the reader, are smart enough to see the "similarity" of these two numbers even though they are not identical in writing ('same' means in their arithmetical property (commutative addition), not their algebraic property (root(2) is the root non-negative of x^2-2)). We hope that the readers will understand the slight carelessness in the use of this terminology.
- If any of the clues seem ambiguous or contradictory, do all the steps that are unambiguous and consistent, and then choose whichever shape you prefer.
