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Sagot :
Sure, let's simplify each expression step by step:
### 1. [tex]\((6x)^3\)[/tex]
First, let's expand the expression:
[tex]\[ (6x)^3 = 6^3 \cdot x^3 \][/tex]
Next, calculate [tex]\(6^3\)[/tex]:
[tex]\[ 6^3 = 6 \cdot 6 \cdot 6 = 216 \][/tex]
So, the simplified expression is:
[tex]\[ (6x)^3 = 216x^3 \][/tex]
### 2. [tex]\((-2x)^2\)[/tex]
First, let's expand the expression:
[tex]\[ (-2x)^2 = (-2)^2 \cdot x^2 \][/tex]
Next, calculate [tex]\((-2)^2\)[/tex]:
[tex]\[ (-2)^2 = (-2) \cdot (-2) = 4 \][/tex]
So, the simplified expression is:
[tex]\[ (-2x)^2 = 4x^2 \][/tex]
### 3. [tex]\(3x^5 \cdot 2x^7 \cdot 7x^7\)[/tex]
First, let's group the constants and the variables together:
[tex]\[ 3 \cdot 2 \cdot 7 \cdot x^5 \cdot x^7 \cdot x^7 \][/tex]
Next, simplify the constants:
[tex]\[ 3 \cdot 2 \cdot 7 = 42 \][/tex]
Then, use the property of exponents ([tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]):
[tex]\[ x^5 \cdot x^7 \cdot x^7 = x^{5+7+7} = x^{19} \][/tex]
So, the simplified expression is:
[tex]\[ 3x^5 \cdot 2x^7 \cdot 7x^7 = 42x^{19} \][/tex]
### 4. [tex]\(7(2x)^2\)[/tex]
First, let's expand the expression inside the parentheses:
[tex]\[ (2x)^2 = (2x) \cdot (2x) = 2^2 \cdot x^2 \][/tex]
Next, calculate [tex]\(2^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
So, the expression inside the parentheses becomes:
[tex]\[ (2x)^2 = 4x^2 \][/tex]
Now, multiply by 7:
[tex]\[ 7 \cdot 4x^2 = 28x^2 \][/tex]
So, the simplified expression is:
[tex]\[ 7(2x)^2 = 28x^2 \][/tex]
### Summary
The simplified expressions are as follows:
[tex]\[ \begin{array}{|l|l|} \hline \text{Original Expression} & \text{Simplified Expression} \\ \hline (6x)^3 & 216x^3 \\ \hline (-2x)^2 & 4x^2 \\ \hline 3x^5 \cdot 2x^7 \cdot 7x^7 & 42x^{19} \\ \hline 7(2x)^2 & 28x^2 \\ \hline \end{array} \][/tex]
### 1. [tex]\((6x)^3\)[/tex]
First, let's expand the expression:
[tex]\[ (6x)^3 = 6^3 \cdot x^3 \][/tex]
Next, calculate [tex]\(6^3\)[/tex]:
[tex]\[ 6^3 = 6 \cdot 6 \cdot 6 = 216 \][/tex]
So, the simplified expression is:
[tex]\[ (6x)^3 = 216x^3 \][/tex]
### 2. [tex]\((-2x)^2\)[/tex]
First, let's expand the expression:
[tex]\[ (-2x)^2 = (-2)^2 \cdot x^2 \][/tex]
Next, calculate [tex]\((-2)^2\)[/tex]:
[tex]\[ (-2)^2 = (-2) \cdot (-2) = 4 \][/tex]
So, the simplified expression is:
[tex]\[ (-2x)^2 = 4x^2 \][/tex]
### 3. [tex]\(3x^5 \cdot 2x^7 \cdot 7x^7\)[/tex]
First, let's group the constants and the variables together:
[tex]\[ 3 \cdot 2 \cdot 7 \cdot x^5 \cdot x^7 \cdot x^7 \][/tex]
Next, simplify the constants:
[tex]\[ 3 \cdot 2 \cdot 7 = 42 \][/tex]
Then, use the property of exponents ([tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]):
[tex]\[ x^5 \cdot x^7 \cdot x^7 = x^{5+7+7} = x^{19} \][/tex]
So, the simplified expression is:
[tex]\[ 3x^5 \cdot 2x^7 \cdot 7x^7 = 42x^{19} \][/tex]
### 4. [tex]\(7(2x)^2\)[/tex]
First, let's expand the expression inside the parentheses:
[tex]\[ (2x)^2 = (2x) \cdot (2x) = 2^2 \cdot x^2 \][/tex]
Next, calculate [tex]\(2^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
So, the expression inside the parentheses becomes:
[tex]\[ (2x)^2 = 4x^2 \][/tex]
Now, multiply by 7:
[tex]\[ 7 \cdot 4x^2 = 28x^2 \][/tex]
So, the simplified expression is:
[tex]\[ 7(2x)^2 = 28x^2 \][/tex]
### Summary
The simplified expressions are as follows:
[tex]\[ \begin{array}{|l|l|} \hline \text{Original Expression} & \text{Simplified Expression} \\ \hline (6x)^3 & 216x^3 \\ \hline (-2x)^2 & 4x^2 \\ \hline 3x^5 \cdot 2x^7 \cdot 7x^7 & 42x^{19} \\ \hline 7(2x)^2 & 28x^2 \\ \hline \end{array} \][/tex]
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