Learning the Theory on Simplifying Radical Expressions

If the radical expression contains a set of mathematical operations with variables, sometimes as a result of its simplicity it is possible to obtain a relatively simple value, a portion of which can be taken out from under the root. Such a simplification may be also useful in those cases when it is necessary to make calculations in mind, and the number under the sign of the root is too big. In this case there is a need to divide the radical expression in several factors, and in order to leave a part of the expression under the radical sign it is required to obtain an accurate result, and removing it from the full value of the square root results in an infinite decimal fraction.

Brief History on Simplifying Radical Expressions

First tasks associated with the extraction of the square root were found in the writings of the Babylonian mathematicians. These tasks include:

• Applying the Pythagorean Theorem to find the side of a right triangle when the values of the other two sides are known.
• Finding the side of the square, with a known area.

Babylonian mathematicians have developed a special numerical method for extracting the square root. The initial approximation to the root was calculated from the natural number n, nearest to the root. Representing the radical expression in the form: a = n2 + r, we obtain: x0 = n + frac {r} / {2n}, then the iterative refinement process corresponding to Newton’s method was used.

Similar tasks and methods can be found in the ancient Chinese «Mathematics in nine books». The ancient Greeks made an important discovery: sqrt {2} is an irrational number. A detailed study carried out by Theaetetus of Athens, showed that if the root is a natural number and it can’t be entirely extracted, then its value is irrational.

The Greeks have formulated the problem of the doubling cube, which was about building a cube root using a ruler and a compass. The problem was insoluble. Numerical algorithms for extracting the cube root were published by Heron and the Indian mathematician Aryabhata I.

Algorithms on extraction of roots of any of degree were developed by Indian and Islamic mathematicians and were improved in medieval Europe. Nicholas Orem for the first time interpreted the root of the n degree as the exponentiation of frac {1} {n}.

After the Cardano formula appeared, the use of mathematics in imaginary numbers began, which was representing the square roots from negative numbers. Basic techniques on working with complex numbers were developed in the 16th century by Rafael Bombelli, who also proposed an original method for calculating the roots (by means of continued fractions).

The founding of the De Moivre’s formula showed that the extraction of the root of any degree from the complex number is always possible and doesn’t lead to a new type of number.

Complex roots of arbitrary degree at the beginning of the 19th century were deeply studied by Gauss, although the first studies belong to Euler. An extremely important discovery by Galois was the proof of the fact that not all algebraic numbers (roots of polynomials) can be obtained from natural numbers with four arithmetic operations and the extraction of the root.

How to Do the Simplifying Radical Expressions?

The first idea of simplifying radical expressions is about the square numbers. Square numbers are any integers with a square root also an integer. For example, 81 with the square root of 9 = (9 x 9 = 81). To reduce the radical expression, which is a square number, simply remove the root sign and write the number, which is the square root of the number of square.

For example, 121 is a square number, because 11 x 11 = 121. You can simply remove the root sign and write 11 as a response.

To reduce this process, you have to remember the first twelve squares of 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 = 121 x 11, 12 x 12 = 144.

The second method is about the simplification of radicands representing the numbers, from which a cube root is taken. These are integers, the cube root of which is also an integer. For example, the number 27 with the cube root = 3 (3 x 3 x 3 = 27). To reduce the radical expression, which is such a number, simply remove the root sign and write down the number that is the cube root of the radicand.

For example, from 512 we can remove a cube root, because the 8 x 8 x 8 = 512. Thus, the cube root of 512 = 8.

The third method of simplifying radical expressions is about usual radical expressions. In this case you need to reduce it into factors. A couple of factors are two numbers when multiplied together give the original number. For example, 5 and 4 is the pair of multipliers of 20. To write the usual radical expression into factors, write down all the factors of this number (or as many as you can imagine, if this number is large) and find among them the square number.

For example, the factors of 45: 1, 3, 5, 9, 15 and 45. 9 is the multiplier of 45 (9 x 5 = 45) and it is also a square number.

Now take the square numbers from the root. 9 is a square number, because the 3 x 3 = 9. Take out 9 from under the root and write 3 in front of it, leaving 5 under the root. If you put number 3 back under the root, it will be multiplied by itself, which is equal to 9, and this value is multiplied by 5 = 45. Square root 3 from 5 is a simplified form of the root of 45.

The fourth option of simplifying radical expressions is connected with the variable. At first, find a quadratic variable. The square root from a in the second degree will be equal to a. The square root from a in the third degree is decomposed in the square root of the product a in the square by a (when multiplying the degrees are added together, so replace 3 = 2 + 1).

Thus, the square variable in the expression a in cube is the squared a. Take all the variables that are square from under the root sign. Now take the square a and put it out from the root, which is equal to a. The simplified form of the root from a in cube is a in the root a.

The fifth method includes simplifying radical expressions with variables and coefficients, which are square. To do this, simply divide the expression into two parts: first, look for the square coefficients, and then look for the square variables. Then take them out of the root. For example, a square root from 36 x a squared.

36 is a square number because 6 x 6 = 36.

a square is a square variable as when you multiply a by a, equals a squared. Now, when you have found square coefficients and variables, take them out from the root. The square root from 36 x a squared is equal to 6a.

Now let’s see on the variant of simplifying radical expressions with the coefficients and variables that are not square. To do this, divide the expression into two parts: first, look for any square coefficients, and then look for any square variables. Then take out the square variables and coefficients you found from under the root sign. For example, consider the square root of 50 x a cube.

Lay out 50 in factors to find among them a square number. 25 x 2 = 50 and 25 is the number, because 5 x 5 = 25. To lay out the root of 50, put 5 out of the root and leave 2 under the root.

Lay out a in the third degree to find a square variable. a in cube is equal to the product of a in square by a, where a in square is a square variable. Take out a out from under the root sign, and leave a under the root. Thus, the root of a in cube is equal to a root from a.

Now connect the two parts. To do this, multiply the values that you have taken out from under the root sign. Do the same with the expressions remaining under the root. Connect 5 root from 2, and a root from a in the expression: 5 x a root from 2 x a.

Online you can find websites, where you can simplify radicands. You simply enter the radical expression, and as a result you can see a simplified expression.

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