Let's find the value of [tex]\( w \)[/tex] in the equation:
[tex]\[
\frac{1}{(11^4)^7} = 11^w
\][/tex]
We will start by simplifying the left-hand side of the equation step by step.
### Step 1: Simplify the exponent on the left-hand side
First, let's simplify [tex]\((11^4)^7\)[/tex]. We use the property of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(11^4)^7 = 11^{4 \cdot 7}
\][/tex]
Calculate the exponent:
[tex]\[
4 \cdot 7 = 28
\][/tex]
So, we have:
[tex]\[
(11^4)^7 = 11^{28}
\][/tex]
### Step 2: Rewrite the fraction using the simplified exponent
Next, write the fraction [tex]\(\frac{1}{11^{28}}\)[/tex]:
[tex]\[
\frac{1}{11^{28}} = 11^{-28}
\][/tex]
Using the property that [tex]\( \frac{1}{a^n} = a^{-n} \)[/tex], we find that:
[tex]\[
\frac{1}{11^{28}} = 11^{-28}
\][/tex]
### Step 3: Equate the simplified expression to the original equation
We originally had the equation:
[tex]\[
\frac{1}{(11^4)^7} = 11^w
\][/tex]
Now we know that:
[tex]\[
\frac{1}{11^{28}} = 11^{-28}
\][/tex]
Substituting this into the original equation, we get:
[tex]\[
11^{-28} = 11^w
\][/tex]
### Step 4: Solve for [tex]\( w \)[/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
-28 = w
\][/tex]
Thus, the value of [tex]\( w \)[/tex] is:
[tex]\[
w = -28
\][/tex]
Therefore, [tex]\( w \)[/tex] in the equation [tex]\(\frac{1}{(11^4)^7} = 11^w\)[/tex] is [tex]\(\boxed{-28}\)[/tex].
