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Sagot :
To determine which expressions are equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex], we need to understand the properties of exponents. Specifically, we should use the property that states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex] where [tex]\( a \)[/tex] is a non-zero constant and [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are exponents. Let's analyze each option step by step:
First, given the expression:
[tex]\[ 7^3 \cdot 7^x \][/tex]
Using the exponent property mentioned, we can combine the terms:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Now let's look at each option:
Option A: [tex]\( 7^{3 x} \)[/tex]
This expression represents a different computation because it means raising 7 to the power of [tex]\( 3x \)[/tex], not [tex]\( 3 + x \)[/tex]. It is not equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex].
Option B: [tex]\( 343 \cdot 7^x \)[/tex]
Let's evaluate [tex]\( 343 \)[/tex]:
[tex]\[ 343 = 7^3 \][/tex]
So, replacing [tex]\( 343 \)[/tex] with [tex]\( 7^3 \)[/tex] in the expression:
[tex]\[ 343 \cdot 7^x = 7^3 \cdot 7^x \][/tex]
Using the exponent property:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Therefore, option B is equivalent to the given expression.
Option C: [tex]\( (7 \cdot x)^3 \)[/tex]
This expression implies raising the product of 7 and [tex]\( x \)[/tex] to the power of 3, which is:
[tex]\[ (7 \cdot x)^3 = 7^3 \cdot x^3 \][/tex]
This is not equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex].
Option D: [tex]\( 7^{3 + x} \)[/tex]
This expression is already in the form we derived:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Thus, option D is equivalent to the given expression.
Option E: [tex]\( 7^{3 - x} \)[/tex]
This expression means [tex]\( 7 \)[/tex] raised to the power of [tex]\( 3 - x \)[/tex], which is different from [tex]\( 7^{3 + x} \)[/tex]. Thus, it is not equivalent.
Option F: [tex]\( 49^{3 x} \)[/tex]
To analyze this option, let's remember that [tex]\( 49 = 7^2 \)[/tex]. Then:
[tex]\[ 49^{3 x} = (7^2)^{3 x} = 7^{2 \cdot 3 x} = 7^{6 x} \][/tex]
This is not the same as [tex]\( 7^{3 + x} \)[/tex]. Thus, it is not equivalent.
Conclusion:
The expressions equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex] are:
- Option B: [tex]\( 343 \cdot 7^x \)[/tex]
- Option D: [tex]\( 7^{3 + x} \)[/tex]
Therefore:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
First, given the expression:
[tex]\[ 7^3 \cdot 7^x \][/tex]
Using the exponent property mentioned, we can combine the terms:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Now let's look at each option:
Option A: [tex]\( 7^{3 x} \)[/tex]
This expression represents a different computation because it means raising 7 to the power of [tex]\( 3x \)[/tex], not [tex]\( 3 + x \)[/tex]. It is not equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex].
Option B: [tex]\( 343 \cdot 7^x \)[/tex]
Let's evaluate [tex]\( 343 \)[/tex]:
[tex]\[ 343 = 7^3 \][/tex]
So, replacing [tex]\( 343 \)[/tex] with [tex]\( 7^3 \)[/tex] in the expression:
[tex]\[ 343 \cdot 7^x = 7^3 \cdot 7^x \][/tex]
Using the exponent property:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Therefore, option B is equivalent to the given expression.
Option C: [tex]\( (7 \cdot x)^3 \)[/tex]
This expression implies raising the product of 7 and [tex]\( x \)[/tex] to the power of 3, which is:
[tex]\[ (7 \cdot x)^3 = 7^3 \cdot x^3 \][/tex]
This is not equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex].
Option D: [tex]\( 7^{3 + x} \)[/tex]
This expression is already in the form we derived:
[tex]\[ 7^3 \cdot 7^x = 7^{3 + x} \][/tex]
Thus, option D is equivalent to the given expression.
Option E: [tex]\( 7^{3 - x} \)[/tex]
This expression means [tex]\( 7 \)[/tex] raised to the power of [tex]\( 3 - x \)[/tex], which is different from [tex]\( 7^{3 + x} \)[/tex]. Thus, it is not equivalent.
Option F: [tex]\( 49^{3 x} \)[/tex]
To analyze this option, let's remember that [tex]\( 49 = 7^2 \)[/tex]. Then:
[tex]\[ 49^{3 x} = (7^2)^{3 x} = 7^{2 \cdot 3 x} = 7^{6 x} \][/tex]
This is not the same as [tex]\( 7^{3 + x} \)[/tex]. Thus, it is not equivalent.
Conclusion:
The expressions equivalent to [tex]\( 7^3 \cdot 7^x \)[/tex] are:
- Option B: [tex]\( 343 \cdot 7^x \)[/tex]
- Option D: [tex]\( 7^{3 + x} \)[/tex]
Therefore:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
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