To simplify the expression [tex]\( 7^{-\frac{5}{6}} \cdot 7^{-\frac{7}{6}} \)[/tex], we can use the properties of exponents. Specifically, we use the property that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Here, our base [tex]\( a \)[/tex] is 7, and the exponents are [tex]\(-\frac{5}{6}\)[/tex] and [tex]\(-\frac{7}{6}\)[/tex]. We add these exponents together to combine them into a single exponent:
[tex]\[ -\frac{5}{6} + -\frac{7}{6} = -\left(\frac{5}{6} + \frac{7}{6}\right) = -\frac{5 + 7}{6} = -\frac{12}{6} = -2 \][/tex]
This means:
[tex]\[ 7^{-\frac{5}{6}} \cdot 7^{-\frac{7}{6}} = 7^{-2} \][/tex]
Next, we simplify [tex]\( 7^{-2} \)[/tex]. The property of negative exponents states:
[tex]\[ a^{-m} = \frac{1}{a^m} \][/tex]
Using this property, we get:
[tex]\[ 7^{-2} = \frac{1}{7^2} \][/tex]
We know that [tex]\( 7^2 = 49 \)[/tex], so:
[tex]\[ 7^{-2} = \frac{1}{49} \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{\frac{1}{49}} \][/tex]
