Category: Math Equation

1. The whole powers

In first and second, you studied the positive integer powers of natural numbers then of integers and finally of fractions and decimals. Until then the powers were positive.

If we can say, the powers have been “invented” in order to simplify the multiplications.

Instead of writing 2.2.2, write 2³.

In third , we move to the higher speed: Arrive the whole powers, so positive or negative while remaining whole. In future years, powers may be fractional.

Note : The negative exponent tells me that I must take the opposite of the designated expression. Thus for example the exponent -3 tells me that I must take the opposite of the expression and then raise this new expression to the cube. A negative power never indicates that the expression considered is negative but simply that we must go to the opposite!

Review the definitions, properties on page 58 and the exercises that you solved on page 59

Small additional explanations : explanations_puissances

Obviously review all the exercises done in class. Rewrite them in the draft, check the answers by comparing them to the answers in the notebook. (That supposes a notebook in order and neat!)

Correction of some exercises on the powers of pages 66 and 67: corr_ex_pages_66_et_67

Complementary exercises :   intro_puissances  and the correction:

If you still want other exercises on the powers, go see on the internet. Just introduce ‘exercises on the powers’ and many corrected exercises will be proposed.

To have relative to the powers:  must_exhaustances   and the correction: 

2.The square roots.

Introduction: see the sheets distributed in class:

intro_racines_carr_es

Before solving the exercises below, review everything that has been done in class, the theoretical notions, the meaning of the symbols, the notations. Also do the exercises done in the notebook. Check the solutions by comparing with the solutions of the notebook.

Some small reminders:

1. How to simplify the radicals: Simplification_of_radicals
2. How to add radicals: Somme_de_radicaux
3. How to multiply radicals: Product_of_rounded_rules
4. Making the denominators rational: Making_the_national_dominators

 

Here are some other ex_radical_1 exercises    and their corrections. Corr_ex_radicaux

and others: ex_radicaux_2

Duty for the: must_role_carr_e_2006_2007

test example: November 8, 2009 test: test_radicaux_09_10

Correction of exercises on page 70 done in class  correction_page_70

These are last year’s equations in second place. I advise you to review in your course last year the exercises that you had solved and the method of resolution.

You can also do all exercises 1 to 7 on page 38 and 86 to 89 on page 46.

Click on the link to find the solutions.

How to solve an equation that seems more complicated, with denominators, large bars of fraction, “less” in front of these bars of fraction, etc.

Click here, you will find ways to learn .

Here are some other equations …. a little more complicated.

Here are the solutions:

 

So far, we are solving equations that admit a single solution. We can say that they are “determined” equations.

Special equations

1) Impossible equations

Either solve the equation: 0x = 4

Question : Is there a number that multiplies by 0 gives 4 as a product? None, of course! 0 being absorbent element for multiplication! This equation does not admit any solution. We will say that it is an impossible equation.

Here is a series of impossible equations.

0x = -3 0x = 78 0x = -56

2) Indeterminate equations

Either solve the equation: 0x = 0

Question : Is there a number that multiplies by 0 gives 0 as a product? Yes of course ! Any number multiplied by 0 gives 0 as product. Any number is therefore solution of this equation. Here is an equation that admits an infinity of solutions. It’s an indeterminate equation .

 

In summary then

An equation has one solution: determined equation

an equation admits an infinity of solutions: indeterminate equation

An equation admits no solution: impossible equation

 

2. The zero-product equations

Reminder : 0 is an absorbing element for multiplication. This means that if a factor of a product is zero, the product is necessarily zero.

Example : 3.5×2,7x0x3,6 = 0 because one of the factors of this product is zero.

Mathematically, whatever the numbers a and b,

If a = 0 or b = 0 then ab = 0

Reciprocally , we will admit that if a product is zero then at least one of its factors is necessarily zero.

Mathematically, whatever the numbers a and b,

if ab = 0, then a = 0 or b = 0

Example :

Here is a zero product: (x + 2) (x + 3) = 0. This product has two factors. Since the product is zero, it is because one of the two factors of the product is necessarily nil. Let the first factor (x + 2) = 0, or the second factor (x + 3) = 0. The product (x + 2) (x + 3) is therefore zero if x = -2 or x = -3.

The numbers found, -2 and -3 are solutions of the equation (x + 2) (x + 3) = 0.

An equation of this type is called a “zero product equation”.

Let’s generalize :

An equation of the type (ax + b) (bx + c) = 0 is called a zero product equation.

The solutions of such an equation are the solutions of each of the equations ax + b = 0 and cx + d = 0

(See the little summary at the bottom of page 32.)

How to reduce an equation to a zero product equation?