The ability to work with numerical expressions containing a square root is necessary for the successful solution of a number of problems from the OGE and the USE. In these exams, a basic understanding of what root extraction is and how it is done in practice is usually sufficient.
Definition
The nth root of a number X is a number x for which the equality is true: xn =X.
Finding the value of an expression with a root means finding x given X and n.
The square root or, which is the same, the second root of X - the number x for which the equality is satisfied: x2 =X.
Designation: ∛X. Here 3 is the degree of the root, X is the root expression. The sign '√' is often called a radical.
If the number above the root does not indicate the degree, then the default is the degree of 2.
In a school course for even degrees, negative roots and radical expressions are usually not considered. For example, there is no√-2, and for the expression √4, the correct answer is 2, despite the fact that (-2)2 also equals 4.
Rationality and irrationality of roots
The simplest possible task with a root is to find the value of an expression or test it for rationality.
For example, calculate the values √25; ∛8; ∛-125:
- √25=5 because 52 =25;
- ∛8=2 because 23 =8;
- ∛ - 125=-5 since (-5)3 =-125.
The answers in the given examples are rational numbers.
When working with expressions that do not contain literal constants and variables, it is recommended to always perform such a check using the inverse operation of raising to a natural power. Finding the number x to the nth power is equivalent to calculating the product of n factors of x.
There are many expressions with a root, the value of which is irrational, that is, written as an infinite non-periodic fraction.
By definition, rationals are those that can be expressed as a common fraction, and irrationals are all other real numbers.
These include √24, √0, 1, √101.
If the problem book says: find the value of the expression with a root of 2, 3, 5, 6, 7, etc., that is, from those natural numbers that are not contained in the table of squares, then the correct answer is √ 2 may be present (unless otherwise stated).
Assessing
In problems withan open answer, if it is impossible to find the value of an expression with a root and write it as a rational number, the result should be left as a radical.
Some assignments may require evaluation. For example, compare 6 and √37. The solution requires squaring both numbers and comparing the results. Of two numbers, the one whose square is greater is greater. This rule works for all positive numbers:
- 62 =36;
- 372 =37;
- 37 >36;
- means √37 > 6.
In the same way, problems are solved in which several numbers must be arranged in ascending or descending order.
Example: Arrange 5, √6, √48, √√64 in ascending order.
After squaring, we have: 25, 6, 48, √64. One could square all the numbers again to compare them with √64, but it equals the rational number 8. 6 < 8 < 25 < 48, so the solution is: 48.
Simplifying the expression
It happens that it is impossible to find the value of an expression with a root, so it must be simplified. The following formula helps with this:
√ab=√a√b.
The root of the product of two numbers is equal to the product of their roots. This operation will also require the ability to factorize a number.
At the initial stage, to speed up the work, it is recommended to have a table of prime numbers and squares at hand. These tables with frequentuse in the future will be remembered.
For example, √242 is an irrational number, you can convert it like this:
- 242=2 × 121;
- √242=√(2 × 121);
- √2 × √121=√2 × 11.
Usually the result is written as 11√2 (read: eleven roots out of two).
If it is difficult to immediately see which two factors a number needs to be decomposed into so that a natural root can be extracted from one of them, you can use the full decomposition into prime factors. If the same prime number occurs twice in the expansion, it is taken out of the root sign. When there are many factors, you can extract the root in several steps.
Example: √2400=√(2 × 2 × 2 × 2 × 2 × 3 × 5 × 5). The number 2 occurs in the expansion 2 times (in fact, more than twice, but we are still interested in the first two occurrences in the expansion).
We take it out from under the root sign:
√(2 × 2 × 2 × 2 × 2 × 3 × 5 × 5)=2√(2 × 2 × 2 × 3 × 5 × 5).
Repeat the same action:
2√(2 × 2 × 2 × 3 × 5 × 5)=2 × 2√(2 × 3 × 5 × 5).
In the remaining radical expression, 2 and 3 occur once, so it remains to take out the factor 5:
2 × 2√(2 × 3 × 5 × 5)=5 × 2 × 2√(2 × 3);
and perform arithmetic operations:
5 × 2 × 2√(2 × 3)=20√6.
So, we get √2400=20√6.
If the task does not explicitly state: "find the value of the expression with a square root", then the choice,in what form to leave the answer (whether to extract the root from under the radical) remains with the student and may depend on the problem being solved.
At first, high requirements are placed on the design of tasks, the calculation, including oral or written, without the use of technical means.
Only after a good mastery of the rules for working with irrational numerical expressions, it makes sense to move on to more difficult literal expressions and to solving irrational equations and calculating the range of possible values of the expression under the radical.
Students encounter this type of problem at the Unified State Exam in mathematics, as well as in the first year of specialized universities when studying mathematical analysis and related disciplines.
