Sure! Let's simplify the given expression step-by-step:
Given expression:
[tex]\[ \left(\frac{10 x^4}{5 x^6}\right)^1 \][/tex]
First, take care of the constants inside the fraction:
[tex]\[ \frac{10}{5} = 2 \][/tex]
So the expression simplifies to:
[tex]\[ \left(2 \frac{x^4}{x^6}\right)^1 \][/tex]
Next, simplify the exponents using the property of exponents \(\frac{x^a}{x^b} = x^{a - b}\):
[tex]\[ \frac{x^4}{x^6} = x^{4 - 6} = x^{-2} \][/tex]
Substituting back:
[tex]\[ \left(2 x^{-2}\right)^1 \][/tex]
Given that any expression raised to the power of 1 remains the same:
[tex]\[ 2 x^{-2} \][/tex]
So the simplified form of the given expression is:
[tex]\[ 2 x^{-2} \][/tex]
If you prefer having only positive exponents, you can rewrite \(x^{-2}\) as \(\frac{1}{x^2}\):
[tex]\[ 2 \cdot \frac{1}{x^2} = \frac{2}{x^2} \][/tex]
Thus, the simplified expression can also be written as:
[tex]\[ \frac{2}{x^2} \][/tex]
In conclusion, the simplified expression is:
[tex]\[ 2 x^{-2} \text{ or } \frac{2}{x^2} \][/tex]
