Basic level 1 option 2.

In task No. 2 of the Unified State Examination in mathematics, it is necessary to demonstrate knowledge of working with power expressions.

Theory for task No. 2

The rules for handling degrees can be presented as follows:

In addition, you should remember about operations with fractions:

Now you can move on to analyzing typical options! πŸ™‚

Analysis of typical options for tasks No. 2 of the Unified State Exam in basic level mathematics

First version of the task

Find the meaning of the expression

Execution algorithm:
  1. Express a number with a negative exponent as a proper fraction.
  2. Perform the first multiplication.
  3. Represent powers of numbers as prime numbers, replacing powers by multiplication.
  4. Perform multiplication.
  5. Perform addition.
Solution:

That is: 10 -1 = 1/10 1 = 1/10

Let's perform the first multiplication, that is, multiplying a whole number by a proper fraction. To do this, multiply the numerator of the fraction by a whole number, and leave the denominator unchanged.

9 1/10 = (9 1)/10 = 9/10

The first power of a number is always the number itself.

The second power of a number is a number multiplied by itself.

10 2 = 10 10 = 100

Answer: 560.9

Second version of the task

Find the meaning of the expression

Execution algorithm:
  1. Represent the first power of a number as an integer.
  2. Represent negative powers of numbers as proper fractions.
  3. Perform multiplication of integers.
  4. Multiply whole numbers by proper fractions.
  5. Perform addition.
Solution:

The first power of a number is always the number itself. (10 1 = 10)

To represent a negative power of a number as an ordinary fraction, you need to divide 1 by this number, but to a positive power.

10 -1 = 1/10 1 = 1/10

10 -2 = 1/10 2 = 1/(10 10) = 1/100

Let's multiply integers.

3 10 1 = 3 10 = 30

Let's multiply whole numbers by proper fractions.

4 10 -2 = 4 1/100 = (4 1)/100 = 4/100

2 10 -1 = 2 1/10 = (2 1)/10 = 2/10

Let us calculate the value of the expression, taking into account that

Answer: 30.24

Third version of the task

Find the meaning of the expression

Execution algorithm:
  1. Represent powers of numbers in the form of multiplication and calculate the value of powers of numbers.
  2. Perform multiplication.
  3. Perform addition.
Solution:

Let's represent powers of numbers in the form of multiplication. In order to represent the power of a number in the form of multiplication, you need to multiply this number by itself as many times as it is contained in the exponent.

2 4 = 2 2 2 2 = 16

2 3 = 2 2 2 = 8

Let's do the multiplication:

4 2 4 = 4 16 = 64

3 2 3 = 3 8 = 24

Let's calculate the value of the expression:

Fourth version of the task

Find the meaning of the expression

Execution algorithm:
  1. Perform the action in parentheses.
  2. Perform multiplication.
Solution:

Let us represent the power of a number in such a way that we can take the common factor out of the bracket.

3 4 3 + 2 4 4 = 4 3 (3 + 2 4)

Let's perform the action in parentheses.

(3 + 2 4) = (3 + 8) = 11

4 3 = 4 4 4 = 64

Let us calculate the value of the expression, taking into account that

Fifth version of the task

Find the meaning of the expression

Execution algorithm:
  1. Let us represent the power of a number in such a way that we can take the common factor out of the bracket.
  2. Place the common factor out of brackets.
  3. Perform the action in parentheses.
  4. Represent the power of a number as a multiplication and calculate the value of the power of the number.
  5. Perform multiplication.
Solution:

Let us represent the power of a number in such a way that we can take the common factor out of the bracket.

Let's take the common factor out of brackets

2 5 3 + 3 5 2 = 5 2 (2 5 + 3)

Let's perform the action in parentheses.

(2 5 + 3) = (10 + 3) = 13

Let's represent the power of a number in the form of multiplication. In order to represent the power of a number in the form of multiplication, you need to multiply this number by itself as many times as it is contained in the exponent.

5 2 = 5 5 = 25

Let us calculate the value of the expression, taking into account that

We perform multiplication in a column, we have:

Option for the second task from the Unified State Exam 2017 (1)

Find the meaning of the expression:

Solution:

In this task, it is more convenient to bring the values ​​to a more familiar form, namely, write the numbers in the numerator and denominator in standard form:

After this, you can divide 24 by 6, the result is 4.

Ten to the fourth power when divided by ten to the third power gives ten to the first, or simply ten, so we get:

Option for the second task from the Unified State Exam 2017 (2)

Find the meaning of the expression:

Solution:

In this case, we should note that the number 6 in the denominator is factored into factors 2 and 3 to the power of 5:

After this, you can perform reductions of degrees for two: 6-5 = 1, for three: 8-5 = 3.

Now we cube 3 and multiply by 2, getting 54.

Option for the second task of 2019 (1)

Execution algorithm
  1. Apply to the numerator of holy powers (a x) y = a xy. We get 3 –6.
  2. Apply to fractions of holy powers a x /a y =a x–y.
  3. Raise 3 to the resulting power.
Solution:

(3 –3) 2 /3 –8 = 3 –6 /3 –8 = 3 –6–(–8)) = 3 –6+8 = 3 2 = 9

Option for the second task 2019 (2)

Execution algorithm
  1. We use for the degree in the numerator (14 9) (ab) x =a x b x. Let us decompose 14 into the product of 2 and 7. We obtain the product of powers with bases 2 and 7.
  2. Let's transform the expression into 2 fractions, each of which will contain powers with the same bases.
  3. Apply to fractions of holy powers a x /a y =a x–y.
  4. We find the resulting product.
Solution:

14 9 / 2 7 7 8 = (2 7) 9 / 2 7 7 8 = 2 9 7 9 / 2 7 7 8 = 2 9–7 7 9–8 = 2 2 7 1 = 4 Β·7 = 28

Option for the second task 2019 (3)

Execution algorithm
  1. We take the common factor 5 2 =25 out of brackets.
  2. We multiply the numbers 2 and 5 in brackets. We get 10.
  3. We add 10 and 3 in brackets. We get 13.
  4. We multiply the common factor 25 and 13.
Solution:

2 5 3 +3 5 2 = 5 2 (2 5+3) = 25 (10+3) = 25 13 = 325

Option for the second task 2019 (4)

Execution algorithm
  1. Square it (–1). We get 1, since it is raised to an even power.
  2. Raise (–1) to the 5th power. We get –1, because raising to an odd power occurs.
  3. We perform multiplication operations.
  4. We get the difference of two numbers. We find her.
Solution:

6Β·(–1) 2 +4Β·(–1) 5 = 6Β·1+4Β·(–1) = 6+(–4) = 6–4 = 2

Option for the second task 2019 (5)

Execution algorithm
  1. Let's convert the factors 10 3 and 10 2 into integers.
  2. We find the products by moving the decimal point to the right by the appropriate number of decimal places.
  3. Find the resulting sum.

Unified State Exam 2019. Mathematics. A basic level of. Typical test tasks. 14 task options.

M.: 2019. - 80 p.

The authors of the manual are leading specialists who are directly involved in the development of methodological materials for preparing for the implementation of control measuring materials of the Unified State Exam. The book contains 14 versions of sets of standard test tasks in mathematics, compiled taking into account all the features and requirements of the Unified State Examination in basic level mathematics. The purpose of the manual is to provide readers with information about the structure and content of test measurement materials in mathematics, the degree of difficulty of tasks. The collection contains answers to all test options. In addition, samples of forms used in the Unified State Exam for recording answers and solutions are provided. The manual can be used by teachers to prepare students for the exam in mathematics in the form of the Unified State Exam, as well as by high school students for self-preparation and self-control.

Format: pdf

Size: 3.8 MB

Watch, download:drive.google

CONTENT
Instructions for performing work 4
References 5
Option 1 9
Option 2 14
Option 3 19
Option 4 24
Option 5 29
Option 6 34
Option 7 39
Option 8 44
Option 9 49
Option 10 54
Option 11 59
Option 12 64
Option 13 69
Option 14 73
Replies 78

The examination paper includes 20 tasks.
3 hours (180 minutes) are allotted to complete the work.
Answers to tasks are written according to the samples below in the form of a number or a sequence of numbers. First, write down the answers to the assignments in the answer field in the text of the work, and then transfer them to answer form No. 1 to the right of the number of the corresponding assignment.
If the answer is a sequence of numbers, as in the example below, then write this sequence on answer sheet No. 1 without spaces, commas or other additional characters.
All Unified State Exam forms are filled out in bright black ink. You can use gel, capillary, or fountain pens.
When completing assignments, you can use a draft. Entries in the draft are not taken into account when grading work.
The points you receive for completed tasks are summed up. Try to complete as many tasks as possible and score the most points.

3. Two candidates ran for the post of chairman of the school council. 84 people took part in the voting. The votes between the candidates were distributed in a ratio of 3:4. How many votes did the winner get?
Answer: .
4. The potential energy of a body (in joules) near the Earth’s surface is calculated by the formula E = mgh, where m is the mass of the body (in kilograms), g is the acceleration of gravity (in m/s2), and L is the height (in meters), on which this body is located, relative to
surfaces. Using this formula, find t (in
kilograms) if g = 9.8 m/s2, h = 5 m, and E = 196 J.
Answer: .
5. Find the value of the expression 26 sin 750Β°.
Answer: .