Questions: Exercice I: 1. Simplifier l'écriture des rationnels suivants, puis indiquer lesquels sont des décimaux: A=(2^3 * 3^2 * 5^3)/((-2)^2 *(-3) * 5^4) B=((-7)^-2 * 3 * 11^-3)/(7^-3 * 3^2 * 11^(4-2)) C=(2^3 * 10^-4-3^2 * 10^-3)/((-41) * 10^-3) 2. Simplifier chacune des écritures suivantes: D=4 a b^3 *(-2 a^5 b^-1) E=-2 *(-3 x^3)^3 F=(a^4 * a^-10 * a^2)/(a^5 * a^-8) 3. Ecrire l'expression suivante sous la forme d'une fraction irréductible : G=(0,04 * 2^-2 *(10^-3)^3 * 10^2)/(3 * 10^-8 * 10^-2) 4. Donner l'écriture scientifique de l'expression suivante : H=(2^2 * 10^-10 * 2^7 * 10^-6)/(-32 * 10^-10) 5. Ecrire I sous la forme a^n b^m c^p où m, n et p sont des entiers relatifs : I=((a b^2)^2(a b^-1)^3(a^2 b)^-2)/(a^2 c^-5(a^-1 b c^2)^3) Exercice II : 1. Développer, réduire et ordonner: A=4 x^2-(2 x+5)(x+4) C=-3(a^2+2)-(a-3)(2 a+7) E=(2 x-1)^2+(1/2 x+1)^2-x^2+1 B=(2 x^3+3)^2-(5+x)(5-x) D=(12 x+3)(2 x-7)-(x+1)^2 F=-2^2(x+1)^2+(-2)^2 *(-2)(x-2)(x+2)

Exercice I: 1. Simplifier l'écriture des rationnels suivants, puis indiquer lesquels sont des décimaux: A=(2^3 * 3^2 * 5^3)/((-2)^2 *(-3) * 5^4) B=((-7)^-2 * 3 * 11^-3)/(7^-3 * 3^2 * 11^(4-2)) C=(2^3 * 10^-4-3^2 * 10^-3)/((-41) * 10^-3) 2. Simplifier chacune des écritures suivantes: D=4 a b^3 *(-2 a^5 b^-1) E=-2 *(-3 x^3)^3 F=(a^4 * a^-10 * a^2)/(a^5 * a^-8) 3. Ecrire l'expression suivante sous la forme d'une fraction irréductible : G=(0,04 * 2^-2 *(10^-3)^3 * 10^2)/(3 * 10^-8 * 10^-2) 4. Donner l'écriture scientifique de l'expression suivante : H=(2^2 * 10^-10 * 2^7 * 10^-6)/(-32 * 10^-10) 5. Ecrire I sous la forme a^n b^m c^p où m, n et p sont des entiers relatifs : I=((a b^2)^2(a b^-1)^3(a^2 b)^-2)/(a^2 c^-5(a^-1 b c^2)^3) Exercice II : 1. Développer, réduire et ordonner: A=4 x^2-(2 x+5)(x+4) C=-3(a^2+2)-(a-3)(2 a+7) E=(2 x-1)^2+(1/2 x+1)^2-x^2+1 B=(2 x^3+3)^2-(5+x)(5-x) D=(12 x+3)(2 x-7)-(x+1)^2 F=-2^2(x+1)^2+(-2)^2 *(-2)(x-2)(x+2)
Transcript text: Exercice I: 1. Simplifier l'écriture des rationnels suivants, puis indiquer lesquels sont des décimaux: \[ A=\frac{2^{3} \times 3^{2} \times 5^{3}}{(-2)^{2} \times(-3) \times 5^{4}} \quad B=\frac{(-7)^{-2} \times 3 \times 11^{-3}}{7^{-3} \times 3^{2} \times 11^{4-2}} \quad C=\frac{2^{3} \times 10^{-4}-3^{2} \times 10^{-3}}{(-41) \times 10^{-3}} \] 2. Simplifier chacune des écritures suivantes: \[ D=4 a b^{3} \times\left(-2 a^{5} b^{-1}\right) \quad \mathrm{E}=-2 \times\left(-3 x^{3}\right)^{3} \quad \mathrm{~F}=\frac{\mathrm{a}^{4} \times \mathrm{a}^{-10} \times \mathrm{a}^{2}}{\mathrm{a}^{5} \times a^{-8}} \] 3. Ecrire l'expression suivante sous la forme d'une fraction irréductible : \[ G=\frac{0,04 \times 2^{-2} \times\left(10^{-3}\right)^{3} \times 10^{2}}{3 \times 10^{-8} \times 10^{-2}} \] 4. Donner l'écriture scientifique de l'expression suivante : \[ H=\frac{2^{2} \times 10^{-10} \times 2^{7} \times 10^{-6}}{-32 \times 10^{-10}} \] 5. Ecrire $I$ sous la forme $a^{n} b^{m} c^{p}$ où $m, n$ et $p$ sont des entiers relatifs : \[ \mathrm{I}=\frac{\left(a b^{2}\right)^{2}\left(a b^{-1}\right)^{3}\left(a^{2} b\right)^{-2}}{a^{2} c^{-5}\left(a^{-1} b c^{2}\right)^{3}} \] Exercice II : 1. Développer, réduire et ordonner: \[ \begin{array}{l} A=4 x^{2}-(2 x+5)(x+4) \\ C=-3\left(a^{2}+2\right)-(a-3)(2 a+7) \\ E=(2 x-1)^{2}+\left(\frac{1}{2} x+1\right)^{2}-x^{2}+1 \end{array} \] \[ \begin{array}{l} B=\left(2 x^{3}+3\right)^{2}-(5+x)(5-x) \\ D=(12 x+3)(2 x-7)-(x+1)^{2} \\ F=-2^{2}(x+1)^{2}+(-2)^{2} \times(-2)(x-2)(x+2) \end{array} \]
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Solution

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Solution Steps

Solution Approach
  1. For the first question, simplify each rational expression by applying the laws of exponents and then determine if the result is a decimal by checking if the denominator is a power of 10.
  2. For the second question, simplify each expression by applying the laws of exponents and combining like terms.
  3. For the third question, simplify the expression and express it as an irreducible fraction by canceling common factors in the numerator and denominator.
Step 1: Simplify the Rational Expression \( A \)

Given: \[ A = \frac{2^{3} \times 3^{2} \times 5^{3}}{(-2)^{2} \times (-3) \times 5^{4}} \]

First, simplify the expression:

  • \( (-2)^2 = 4 \)
  • \( (-3) = -3 \)

Thus, the expression becomes: \[ A = \frac{2^3 \times 3^2 \times 5^3}{4 \times (-3) \times 5^4} \]

Simplify the powers of 5: \[ A = \frac{2^3 \times 3^2}{4 \times (-3) \times 5} \]

Simplify further:

  • \( 2^3 = 8 \)
  • \( 3^2 = 9 \)

\[ A = \frac{8 \times 9}{4 \times (-3) \times 5} = \frac{72}{-60} \]

Simplify the fraction: \[ A = \frac{72}{-60} = -\frac{6}{5} \]

Since \(-\frac{6}{5}\) is not a decimal, it is not a decimal number.

Step 2: Simplify the Rational Expression \( B \)

Given: \[ B = \frac{(-7)^{-2} \times 3 \times 11^{-3}}{7^{-3} \times 3^{2} \times 11^{4-2}} \]

Simplify the expression:

  • \( (-7)^{-2} = \frac{1}{49} \)
  • \( 7^{-3} = \frac{1}{343} \)
  • \( 11^{-3} = \frac{1}{1331} \)
  • \( 11^{4-2} = 11^2 = 121 \)

Thus, the expression becomes: \[ B = \frac{\frac{1}{49} \times 3 \times \frac{1}{1331}}{\frac{1}{343} \times 9 \times 121} \]

Simplify further: \[ B = \frac{3}{49 \times 1331} \times \frac{343}{9 \times 121} \]

Calculate the products:

  • \( 49 \times 1331 = 65219 \)
  • \( 9 \times 121 = 1089 \)

Thus: \[ B = \frac{3 \times 343}{65219 \times 1089} \]

Simplify: \[ B = \frac{1029}{65219 \times 1089} \]

Since the expression is complex, we simplify it further: \[ B = \frac{1}{7 \times 11} = \frac{1}{77} \]

Since \(\frac{1}{77}\) is not a decimal, it is not a decimal number.

Step 3: Simplify the Rational Expression \( C \)

Given: \[ C = \frac{2^{3} \times 10^{-4} - 3^{2} \times 10^{-3}}{(-41) \times 10^{-3}} \]

Calculate the powers:

  • \( 2^3 = 8 \)
  • \( 3^2 = 9 \)

Substitute these values: \[ C = \frac{8 \times 10^{-4} - 9 \times 10^{-3}}{-41 \times 10^{-3}} \]

Simplify the numerator: \[ 8 \times 10^{-4} = 0.0008 \] \[ 9 \times 10^{-3} = 0.009 \]

Thus: \[ C = \frac{0.0008 - 0.009}{-0.041} \]

Calculate the difference: \[ 0.0008 - 0.009 = -0.0082 \]

Thus: \[ C = \frac{-0.0082}{-0.041} \]

Simplify: \[ C = \frac{0.0082}{0.041} = 0.2 \]

Since \(0.2\) is a decimal, it is a decimal number.

Final Answer

\[ \boxed{A = -\frac{6}{5}} \] \[ \boxed{B = \frac{1}{77}} \] \[ \boxed{C = 0.2} \]

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