To solve the given problem, we need to find the value of [tex]\( x^{-4} \)[/tex] given that [tex]\((x^5 \times x^{-2})^2 = 64\)[/tex].
First, let's simplify the expression inside the parentheses:
1. [tex]\((x^5 \times x^{-2})^2\)[/tex]
- Using the property of exponents [tex]\( a^m \times a^n = a^{m+n} \)[/tex], we can combine the exponents:
[tex]\[ x^5 \times x^{-2} = x^{5 + (-2)} = x^{3} \][/tex]
2. Now substitute this simplified expression back into the original equation:
[tex]\[ (x^3)^2 = 64 \][/tex]
3. Apply the power of a power property, which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ x^{3 \cdot 2} = x^{6} = 64 \][/tex]
4. To solve for [tex]\( x \)[/tex], we need to take the sixth root of both sides of the equation:
[tex]\[ x = 64^{\frac{1}{6}} \][/tex]
So, [tex]\( x = 2.0 \)[/tex].
Next, we need to find the value of [tex]\( x^{-4} \)[/tex]:
1. Substitute [tex]\( x = 2.0 \)[/tex] into [tex]\( x^{-4} \)[/tex]:
[tex]\[ x^{-4} = (2.0)^{-4} \][/tex]
2. Calculate the value:
[tex]\[ (2.0)^{-4} = \frac{1}{(2.0)^4} = \frac{1}{16} = 0.0625 \][/tex]
Therefore, the value of [tex]\( x^{-4} \)[/tex] is [tex]\( 0.0625 \)[/tex].
