Lesson script for 11th grade on the topic:
“The nth root of a real number. »
The purpose of the lesson: Formation in students of a holistic understanding of the root n-th degree and arithmetic root of the nth degree, formation of computational skills, skills of conscious and rational use of the properties of the root when solving various problems containing a radical. Check the level of students' understanding of the topic's questions.
Subject:create meaningful and organizational conditions for mastering material on the topic “ Numeric and alphabetic expressions » at the level of perception, comprehension and primary memorization; develop the ability to use this information when calculating the nth root of a real number;
Meta-subject: promote the development of computing skills; ability to analyze, compare, generalize, draw conclusions;
Personal: cultivate the ability to express one’s point of view, listen to the answers of others, take part in dialogue, and develop the ability for positive cooperation.
Planned result.
Subject: be able to apply in a real situation the properties of the nth root of a real number when calculating roots and solving equations.
Personal: to develop attentiveness and accuracy in calculations, a demanding attitude towards oneself and one’s work, and to cultivate a sense of mutual assistance.
Lesson type: lesson on studying and initially consolidating new knowledge
Motivation for educational activities:
Eastern wisdom says: “You can lead a horse to water, but you cannot force him to drink.” And it is impossible to force a person to study well if he himself does not try to learn more and does not have the desire to work on his mental development. After all, knowledge is only knowledge when it is acquired through the efforts of one’s thoughts, and not through memory alone.
Our lesson will be held under the motto: “We will conquer any peak if we strive for it.” During the lesson, you and I need to have time to overcome several peaks, and each of you must put all your efforts into conquering these peaks.
“Today we have a lesson in which we must get acquainted with a new concept: “Nth root” and learn how to apply this concept to the transformation of various expressions.
Your goal is to activate your existing knowledge through various forms of work, contribute to the study of the material and get good grades.”
We studied the square root of a real number in 8th grade. The square root is related to a function of the form y=x 2. Guys, do you remember how we calculated square roots, and what properties did it have?
a) individual survey: 
what kind of expression is this
what is called square root
what is called arithmetic square root
list the properties of square root
b) work in pairs: calculate.
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2. Updating knowledge and creating a problem situation: Solve the equation x 4 =1. How can we solve it? (Analytical and graphical). Let's solve it graphically. To do this, in one coordinate system we will construct a graph of the function y = x 4 straight line y = 1 (Fig. 164 a). They intersect at two points: A (-1;1) and B(1;1). Abscissas of points A and B, i.e. x 1 = -1,
x 2 = 1 are the roots of the equation x 4 = 1.
Reasoning in exactly the same way, we find the roots of the equation x 4 =16: Now let’s try to solve the equation x 4 =5; a geometric illustration is shown in Fig. 164 b. It is clear that the equation has two roots x 1 and x 2, and these numbers, as in the two previous cases, are mutually opposite. But for the first two equations the roots were found without difficulty (they could be found without using graphs), but with the equation x 4 = 5 there are problems: from the drawing we cannot indicate the values of the roots, but we can only establish that one root is located to the left point -1, and the second one is to the right of point 1.
x 2 = - (read: “fourth root of five”).
We talked about the equation x 4 = a, where a 0. We could equally well talk about the equation x 4 = a, where a 0, and n is any natural number. For example, solving graphically the equation x 5 = 1, we find x = 1 (Fig. 165); solving the equation x 5 "= 7, we establish that the equation has one root x 1, which is located on the x axis slightly to the right of point 1 (see Fig. 165). For the number x 1, we introduce the notation .
Definition 1. The nth root of a non-negative number a (n = 2, 3,4, 5,...) is a non-negative number that, when raised to the power n, results in the number a.
This number is denoted, the number a is called the radical number, and the number n is the exponent of the root.
If n=2, then they usually don’t say “second root,” but say “square root.” In this case, they don’t write this. This is the special case that you specifically studied in the 8th grade algebra course.
If n = 3, then instead of “third degree root” they often say “cube root”. Your first acquaintance with the cube root also took place in the 8th grade algebra course. We used cube roots in 9th grade algebra.
So, if a ≥0, n= 2,3,4,5,…, then 1) ≥ 0; 2) () n = a.
In general, =b and b n =a are the same relationship between non-negative numbers a and b, but only the second is described in a simpler language (uses simpler symbols) than the first.
The operation of finding the root of a non-negative number is usually called root extraction. This operation is the reverse of raising to the appropriate power. Compare:
Please note again: only positive numbers appear in the table, since this is stipulated in Definition 1. And although, for example, (-6) 6 = 36 is a correct equality, go from it to notation using the square root, i.e. write that it is impossible. By definition, a positive number means = 6 (not -6). In the same way, although 2 4 =16, t (-2) 4 =16, moving to the signs of the roots, we must write = 2 (and at the same time ≠-2).
Sometimes the expression is called a radical (from the Latin word gadix - “root”). In Russian, the term radical is used quite often, for example, “radical changes” - this means “radical changes”. By the way, the very designation of the root is reminiscent of the word gadix: the symbol is a stylized letter r.
The operation of extracting the root is also determined for a negative radical number, but only in the case of an odd root exponent. In other words, the equality (-2) 5 = -32 can be rewritten in equivalent form as =-2. The following definition is used.
Definition 2. An odd root n of a negative number a (n = 3.5,...) is a negative number that, when raised to the power n, results in the number a.
This number, as in Definition 1, is denoted by , the number a is the radical number, and the number n is the exponent of the root.
So, if a , n=,5,7,…, then: 1) 0; 2) () n = a.
Thus, an even root has meaning (i.e., is defined) only for a non-negative radical expression; an odd root makes sense for any radical expression.
5. Primary consolidation of knowledge:
1. Calculate: No. 33.5; 33.6; 33.74 33.8 orally a) ; b) ; V) ; G) .
d) Unlike previous examples, we cannot indicate the exact value of the number. It is only clear that it is greater than 2, but less than 3, since 2 4 = 16 (this is less than 17), and 3 4 = 81 (this more than 17). We note that 24 is much closer to 17 than 34, so there is reason to use the approximate equality sign:
2.
Find the meanings of the following expressions.
Place the corresponding letter next to the example.
A little information about the great scientist. Rene Descartes (1596-1650) French nobleman, mathematician, philosopher, physiologist, thinker. Rene Descartes laid the foundations of analytical geometry and introduced the letter designations x 2, y 3. Everyone knows the Cartesian coordinates that define a function of a variable.
3 . Solve the equations: a) = -2; b) = 1; c) = -4
Solution: a) If = -2, then y = -8. In fact, we must cube both sides of the given equation. We get: 3x+4= - 8; 3x= -12; x = -4. b) Reasoning as in example a), we raise both sides of the equation to the fourth power. We get: x=1.
c) There is no need to raise it to the fourth power; this equation has no solutions. Why? Because, according to definition 1, an even root is a non-negative number.
Several tasks are offered to your attention. When you complete these tasks, you will learn the name and surname of the great mathematician. This scientist was the first to introduce the root sign in 1637.
6. Let's have a little rest.
The class raises its hands - this is “one”.
The head turned - it was “two”.
Hands down, look forward - this is “three”.
Hands turned wider to the sides to “four”
Pressing them with force into your hands is a “high five.”
All the guys need to sit down - it’s “six”.
7. Independent work:
option: option 2:
b) 3-. b)12 -6.
2. Solve the equation: a) x 4 = -16; b) 0.02x 6 -1.28=0; a) x 8 = -3; b)0.3x 9 – 2.4=0;
c) = -2; c)= 2
8. Repetition: Find the root of the equation = - x. If the equation has more than one root, write the answer with the smaller root.
9. Reflection: What did you learn in the lesson? What was interesting? What was difficult?
Degree formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.
Number c is n-th power of a number a When:
Operations with degrees.
1. By multiplying degrees with the same base, their indicators are added:
a m·a n = a m + n .
2. When dividing degrees with the same base, their exponents are subtracted:
3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:
(abc…) n = a n · b n · c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n /b n .
5. Raising a power to a power, the exponents are multiplied:
(a m) n = a m n .
Each formula above is true in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:
3. When raising a root to a power, it is enough to raise the radical number to this power:
4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:
5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:
A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:
Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.
A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.