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**Answer key for math exercises in 2nd grade on calculations and problems**. Develop skills with fractions and powers.

Exercise 1:

Either

a. Show that

first of all

Let be the function , this function is strictly increasing on .

so

thus

and therefore

b. Calculate .

c. Deduce the value of There are two possible values X=2 or X=-2

or

from 1 X<0 , **we deduce that X = – 2 .**

Exercise 2:

Calculate:

Exercise 3:

Exercise 4:

1. Develop and reduce:

a= (5x+1)(2x+3)

b= (4x-5)(7x-1)

c= (2x+5)(7x-3)

d= (-4x-6)(2x-1)

2. Expand and reduce:

e= (5x+1)(2x+3)+(5x+1)(x+2)

f= (4x-5)(7x-1)-(4x-5)(3x+2)

g= (-4x-6)(2x-1)+(2x-3)(8x-11)

h= (x-8)(5+3x)-(x-8)(7-x)

Exercise 5:

Frame when .

then

Exercise 6:

Factor the following expressions:

Exercise 7:

Exercise 8:

1. In what form is an even number written ?

2. In what form is an odd number written?

3. Show that the square of an even number is an even number.

with

**Conclusion: the square of an even number is an even number.**

Exercise 9:

1. Calculate the sum of 5 consecutive integers.

**It’s up to you to do some tests.**

2. Show that the sum of five consecutive integers is a multiple of 5.

Let x be the smallest integer .

Either

**So the sum of 5 consecutive integers is a multiple of 5.**

Exercise 10:

1. Calculate the product of four consecutive integers and add 1.

What do we notice? (Make several attempts)

2x3x4x5+1=121=11²

5x6x7x8+1=1681=41²

6x7x8x9+1=3025=552

2. Show that, for any real x, we have :

Explain the result observed in question 1.

The product of 4 consecutive numbers plus 1 can be written as the square of a number.

Exercise 11:

a. Decompose into the product of prime numbers 220 and 284.

b. Check that 220 and 284 are friendly.

The divisors of 220 are 1;2;4;5;10;11;20;22;44;55;110;220.

The divisors of 284 are 1;2;4;71;142;284

or let’s calculate the sum of the divisors except m :

and

**Conclusion: 220 and 284 are two friendly integers.**

Exercise 12:

Write in interval form :

Exercise 13:

1. Hervé must factor A.

Here’s his answer:

Test the equality obtained by Hervé for

What can we conclude from this?

and if we replace 0 in the original expression, we obtain :

**Conclusion :** Hervé made a mistake when factoring.

2. To factor A, we can think of writing :

Then factor A correctly.

Exercise 14:

Factor each expression by highlighting a common factor.

Exercise 15:

Exercise 16:

Exercise 17:

Exercise 18:

Exercise 19:

Exercise 20:

The golden number is the number .

**Check the following equalities:**

1. .

2. .

Let’s multiply by :

(this is the equality of question 1.)

3.

We know that :

Let’s multiply this equality by :

(because )

so

Exercise 21:

The speed of light is estimated at m/s

and the average distance Earth-Sun at 149 million kilometers.

**Calculate the time needed for a light signal from the Earth to reach the**

**Sun.**

Exercise 22:

For , calculate :

Exercise 23:

Prove that the diagonal of a square of side is .

In the right-angled triangle ABC at B, according to the direct part of the Pythagorean theorem :

(a length is positive or zero)

Exercise 24:

Exercise 25:

Exercise 26:

Exercise 28:

1. For each line, reconstruct the sentence using **Si……then ….** or **…if and only if ….:**

a.

IF IT RAINS THEN I take my umbrella.

IF I in the middle of [AB] THEN AI=BI

IF AND ONLY IF.

IF THEN .

IF THEN ABC is isosceles at A.

Exercise 29:

For n natural numbers, compare the following numbers:

Exercise 30:

Let ,compare the numbers :

Suppose that

So

So

So

So

this being always true since

so

Exercise 31:

1. Complete using the symbols

a.

b.

2. Specify the interval corresponding to :

a.

b.

c.

Exercise 32:

a.

b.

2. Simplify and then write the following fraction in scientific form:

3. Simplify the following entries:

a.

b.

c.

4. Give the prime factor product decomposition of the number .

Exercise 33:

a. Indicate the nature of the following numbers:

: rational : rational

: integer D=1+: irrational

b. Simplify the writing of the following number:

After having consulted

**the answers to these exercises on calculations and problems in grade 2**, you can return to the

**exercises in**grade

**2**.

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