Alright, let's expand the expression [tex]\(\left(x^{10} - 5y^2\right)^2\)[/tex] step-by-step.
We start with the given expression:
[tex]\[ \left(x^{10} - 5 y^2\right)^2 \][/tex]
To expand this expression, we will use the algebraic identity for the square of a binomial, which is:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
where [tex]\( a = x^{10} \)[/tex] and [tex]\( b = 5y^2 \)[/tex].
Using this identity, we have:
[tex]\[ \left(x^{10} - 5 y^2\right)^2 = (x^{10})^2 - 2(x^{10})(5 y^2) + (5 y^2)^2 \][/tex]
Now, let's compute each term separately:
1. The first term is [tex]\((x^{10})^2\)[/tex]:
[tex]\[ (x^{10})^2 = x^{10 \cdot 2} = x^{20} \][/tex]
2. The second term is [tex]\(-2(x^{10})(5 y^2)\)[/tex]:
[tex]\[ -2(x^{10})(5 y^2) = -2 \cdot 5 \cdot x^{10} \cdot y^2 = -10 x^{10} y^2 \][/tex]
3. The third term is [tex]\((5 y^2)^2\)[/tex]:
[tex]\[ (5 y^2)^2 = 5^2 (y^2)^2 = 25 (y^2 \cdot y^2) = 25 y^{2 \cdot 2} = 25 y^4 \][/tex]
Combining these three terms together, we obtain the expanded form:
[tex]\[ \left(x^{10} - 5 y^2\right)^2 = x^{20} - 10 x^{10} y^2 + 25 y^4 \][/tex]
So, the fully expanded form of [tex]\(\left(x^{10} - 5 y^2\right)^2\)[/tex] is:
[tex]\[ x^{20} - 10 x^{10} y^2 + 25 y^4 \][/tex]
