To determine the degree of the polynomial \(2^{10} y^4 + 3 y^9 - 2 y^3 x^2 + 5^3 x^7\), we need to consider each term individually and find its degree. The degree of a term in a polynomial is the sum of the exponents of all variables in that term. Let's analyze each term step-by-step.
1. First term: \(2^{10} y^4\)
- The term is \(2^{10} y^4\).
- The only variable is \(y\) with exponent 4.
- Therefore, the degree of this term is \(4\).
2. Second term: \(3 y^9\)
- The term is \(3 y^9\).
- The only variable is \(y\) with exponent 9.
- Therefore, the degree of this term is \(9\).
3. Third term: \(-2 y^3 x^2\)
- The term is \(-2 y^3 x^2\).
- There are two variables: \(y\) with exponent 3 and \(x\) with exponent 2.
- The degree of this term is the sum of the exponents: \(3 + 2 = 5\).
4. Fourth term: \(5^3 x^7\)
- The term is \(5^3 x^7\).
- The only variable is \(x\) with exponent 7.
- Therefore, the degree of this term is \(7\).
After determining the degree of each term:
- \(2^{10} y^4\) has degree 4,
- \(3 y^9\) has degree 9,
- \(-2 y^3 x^2\) has degree 5,
- \(5^3 x^7\) has degree 7.
The degree of the polynomial is the highest degree among these terms. The degrees we found are \(4\), \(9\), \(5\), and \(7\). The highest degree is \(9\).
Therefore, the degree of the polynomial [tex]\(2^{10} y^4 + 3 y^9 - 2 y^3 x^2 + 5^3 x^7\)[/tex] is [tex]\(\boxed{9}\)[/tex].
