To solve for [tex]\( x^{-4} \)[/tex], let's start with the given equation:
[tex]\[ \left(x^5 \times x^{-2}\right)^2 = 64 \][/tex]
First, simplify the expression inside the parentheses. Remember that when you multiply powers of the same base, you add the exponents:
[tex]\[ x^5 \times x^{-2} = x^{5 + (-2)} = x^3 \][/tex]
So the given equation simplifies to:
[tex]\[ (x^3)^2 = 64 \][/tex]
Next, use the property of exponents where [tex]\((a^m)^n = a^{m \times n}\)[/tex]:
[tex]\[ x^{3 \times 2} = x^6 \][/tex]
Thus, the equation becomes:
[tex]\[ x^6 = 64 \][/tex]
To solve for [tex]\( x \)[/tex], take the sixth root of both sides:
[tex]\[ x = \sqrt[6]{64} \][/tex]
We know that [tex]\( 64 \)[/tex] can be written as [tex]\( 2^6 \)[/tex]:
[tex]\[ x = \sqrt[6]{2^6} = 2 \][/tex]
Now, we need to find [tex]\( x^{-4} \)[/tex]. Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[ x^{-4} = 2^{-4} \][/tex]
Recall that [tex]\( 2^{-4} = \frac{1}{2^4} \)[/tex], and since [tex]\( 2^4 = 16 \)[/tex]:
[tex]\[ x^{-4} = \frac{1}{16} = 0.0625 \][/tex]
So, the value of [tex]\( x^{-4} \)[/tex] is:
[tex]\[ \boxed{0.0625} \][/tex]
