Let's solve the equation [tex]\( 7^x \cdot 3^x \cdot 2^x = 1764 \)[/tex].
1. Combine the exponents:
Notice that the left-hand side of the equation can be rewritten by combining the exponents:
[tex]\[
(7 \cdot 3 \cdot 2)^x = 1764
\][/tex]
Simplify inside the parentheses:
[tex]\[
(42)^x = 1764
\][/tex]
2. Rewrite 1764 as a power of 42:
We need to express 1764 as a power of 42. Let's check if 1764 is indeed a power of 42. We will perform the calculations to see if it matches:
[tex]\[
42^2 = 42 \times 42 = 1764
\][/tex]
Since [tex]\( 42^2 = 1764 \)[/tex], we can rewrite the equation as:
[tex]\[
(42)^x = 42^2
\][/tex]
3. Equate the exponents:
Since the bases (42) are the same, their exponents must be equal:
[tex]\[
x = 2
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{2} \)[/tex].
