To determine the missing term [tex]\( x \)[/tex] in the equation
[tex]\[
\left(\frac{3}{6}\right)^{x} = \frac{243}{7776},
\][/tex]
you can follow the steps below:
1. Simplify the base fraction:
[tex]\[
\frac{3}{6} = \frac{1}{2}
\][/tex]
Thus, the equation becomes:
[tex]\[
\left(\frac{1}{2}\right)^{x} = \frac{243}{7776}
\][/tex]
2. Simplify the right-hand side fraction:
[tex]\[
\frac{243}{7776}
\][/tex]
We know that:
[tex]\[
243 = 3^5 \quad \text{and} \quad 7776 = 6^5
\][/tex]
Since [tex]\( 6 = 2 \times 3 \)[/tex], we have:
[tex]\[
7776 = (2 \times 3)^5 = 2^5 \times 3^5
\][/tex]
Therefore:
[tex]\[
\frac{243}{7776} = \frac{3^5}{2^5 \times 3^5} = \frac{1}{2^5} = \left(\frac{1}{2}\right)^5
\][/tex]
3. Rewrite the equation with the simplified terms:
[tex]\[
\left(\frac{1}{2}\right)^{x} = \left(\frac{1}{2}\right)^5
\][/tex]
4. Equate the exponents:
If the bases are equal, the exponents must be equal:
[tex]\[
x = 5
\][/tex]
Thus, the value of the missing term is [tex]\( x = 5 \)[/tex], which corresponds to option C. 5.
