To determine the values of \( a \), \( b \), and \( c \) from the given equations, let's solve each equation one by one.
1. Determine the value of \( a \) in the equation \((xa)^6 = \frac{1}{x^{30}}\)
First, we rewrite the left side of the equation:
\[ (xa)^6 = x^6 a^6 \]
Now, set the two sides equal to each other:
\[ x^6 a^6 = \frac{1}{x^{30}} \]
Recall that:
\[ \frac{1}{x^{30}} = x^{-30} \]
So we have:
\[ x^6 a^6 = x^{-30} \]
This means that the exponents of \( x \) on both sides must be equal:
\[ 6 + \log_x(a^6) = -30 \]
From \( x^6 a^6 = x^{-30} \), we can also notice that \( a^6 \) itself must have a term corresponding to \( x^{-30} \) so we only set:
\[ 6 + \log_x(a^6) = -30 \]
Thus,
\[ 6 = -30 \Rightarrow a^6 = x^{-36} \]
Therefore, from \( 6 + log_x(a^6) = -30 \), we re-focus and simply rearrange to:
Hence, \( 6 + \log_x(a) = -30 \)
This implies:
\[ \log_x(a) = -30 - 6 = -36 \] Thus, we can write:
\[ a = x^{-36} \]
Thus, since we need only the constant value of the exponent:
\[ a = -36 \]
2. Determine the value of \( b \) in the equation \((x^{-7})^{-4} = x^b\)
Apply the power of a power property:
\[ (x^{-7})^{-4} = x^{-7 \cdot -4} = x^{28} \]
Now, equate the exponents:
\[ b = 28 \]
3. Determine the value of \( c \) in the equation \((x^{-2})^c = x^{22}\)
Again, applying the power of a power property, we can rewrite the left-hand side:
\[ (x^{-2})^c = x^{-2c} \]
Now, set the exponents equal:
\[ -2c = 22 \]
To solve for \( c \), divide both sides by -2:
\[ c = \frac{22}{-2} = -11 \]
Summary of Values
- \( a = -36 \)
- \( b = 28 \)
- \( c = -11 \)
Thus,
\[ \boxed{a = -36}, \quad \boxed{b = 28}, \quad \boxed{c = -11} \]