The ability to work with numerical expressions containing a square root is necessary for the successful solution of a number of problems from the unified state exam and the unified state exam. As a rule, in these exams, a basic understanding of what root extraction is and how it is practiced is sufficient.
Definition
A root of degree n from X is a number x for which the equality holds: x n = X.
To find the value of an expression with a root means to find x for known X and n.
The square root or, equivalently, the second-degree root of X is the number x for which the equality holds: x 2 = X.
Designation: ∛X. Here 3 is the root degree, X is the radical expression. The sign '√' is often called a radical.
If there is no number indicating the degree above the root, then the default is degree 2.
In the school course, for even degrees, negative roots and radical expressions are usually not considered. For example, √-2 does not exist, and for the expression √4, 2 is considered the correct answer, despite the fact that (-2) 2 is also 4.
Rationality and irrationality of the roots
The simplest possible task with the root is to find the value of the expression or check it for rationality.
For example, calculate √25; ∛8; ∛-125:
- √25 = 5, since 5 2 = 25;
- ∛8 = 2, since 2 3 = 8;
- ∛ - 125 = -5, since (-5) 3 = -125.
The answers in the examples are rational numbers.
When working with expressions that do not contain literal constants and variables, it is recommended that you always perform a similar check using the inverse operation of raising to a natural power. Finding the number x to the nth power is equivalent to computing the product of n factors x.
There are many expressions with a root, the meaning of which is irrational, that is, it is written in the form of an infinite non-periodic fraction.
By definition, rationals are those that can be expressed with an ordinary fraction, while irrational ones are all other real numbers.
These include √24, √0.1, √101.
If the problem book says: find the value of an 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 √2 may be present in the correct answer (when not specified otherwise).
Assessment
In problems with an open answer, if it is impossible to find the value of an expression with a root and write it down with a rational number, the result should be left in the form of a radical.
Some tasks may require an assessment. For example, compare 6 and √37. To solve, you need to square both numbers and compare the results. Of the two numbers, the larger is whose square is the larger. This rule works for all positive numbers:
- 6 2 = 36;
- 37 2 = 37;
- 37> 36;
- hence √37> 6.
In the same way, problems are solved in which several numbers must be arranged in ascending or descending order.
Example: arrange in ascending order 5, √6, √48, √√64.
After squaring we have: 25, 6, 48, √64. It would be possible to square all the numbers again in order to compare them with √64, but it is equal to the rational number 8. 6 <8 <25 <48, so the solution is: √6 <√√64 <5 <√ 48.
Simplification of Expression
It happens that it is impossible to find the value of an expression with a root, so it needs to be simplified. The following formula helps in 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 factor the number.
At the initial stage, to speed up the work, it is recommended to have at hand a table of primes and squares. These tables with frequent use will be remembered in the future.
For example, √242 is an irrational number, you can convert it like this:
- 242 = 2 × 121;
- √242 = √ (2 × 121);
- √2 × √121 = √2 × 11.
Typically, the result is written as 11√2 (read: eleven of the two roots).
If it is difficult to see at once on which two factors you need to factor the number so that a natural root is extracted from one of them, you can use the full factorization. If the same prime number in the expansion occurs twice, it is taken out for the root sign. When there are many factors, you can extract the root in several actions.
Example: √2400 = √ (2 × 2 × 2 × 2 × 2 × 3 × 5 × 5). The number 2 was found in the decomposition 2 times (in fact, more than two times, but so far we are interested in the first two occurrences of the decomposition).
We take it out from under the sign of the root:
√ (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 root expression, 2 and 3 occur once, so it remains to factor out 5:
2 × 2√ (2 × 3 × 5 × 5) = 5 × 2 × 2√ (2 × 3);
and perform arithmetic:
5 × 2 × 2√ (2 × 3) = 20√6.
So, we get √2400 = 20√6.
If the assignment does not explicitly state: “find the value of the expression with a square root”, then the choice in which 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 were placed on the execution of tasks, the calculation, including oral or written, without the use of technical means.
Only after a good understanding of the rules for working with irrational numerical expressions does it make sense to move on to more difficult literal expressions and to solve irrational equations and calculate the range of possible values of the expression under the radical.
Students encounter this type of tasks at the USE in mathematics, as well as in the first year of specialized universities in the study of mathematical analysis and related disciplines.