To find the value of [tex]\(\frac{x^{m+n} \times y^{n-m}}{x^{m-n} \times y^{n+m}}\)[/tex] given [tex]\(x=2, y=4, m=-1,\)[/tex] and [tex]\(n=3\)[/tex], we can proceed step-by-step:
1. Evaluate the exponents in the numerator and denominator:
- [tex]\(m+n = -1 + 3 = 2\)[/tex]
- [tex]\(n-m = 3 - (-1) = 3 + 1 = 4\)[/tex]
- [tex]\(m-n = -1 - 3 = -4\)[/tex]
- [tex]\(n+m = 3 - 1 = 2\)[/tex]
2. Substitute the exponents back into the expression:
[tex]\[
\frac{x^{m+n} \times y^{n-m}}{x^{m-n} \times y^{n+m}} = \frac{x^2 \times y^4}{x^{-4} \times y^2}
\][/tex]
3. Substitute the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- [tex]\(x=2\)[/tex]
- [tex]\(y=4\)[/tex]
[tex]\[
\frac{2^2 \times 4^4}{2^{-4} \times 4^2}
\][/tex]
4. Calculate each term:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(4^4 = (2^2)^4 = 2^8 = 256\)[/tex]
- [tex]\(2^{-4} = \frac{1}{2^4} = \frac{1}{16}\)[/tex]
- [tex]\(4^2 = (2^2)^2 = 2^4 = 16\)[/tex]
5. Substitute these results back into the expression:
[tex]\[
\frac{4 \times 256}{\frac{1}{16} \times 16}
\][/tex]
6. Simplify the denominator:
- [tex]\(\frac{1}{16} \times 16 = \frac{16}{16} = 1\)[/tex]
7. Thus, the expression reduces to:
[tex]\[
\frac{4 \times 256}{1} = 4 \times 256 = 1024
\][/tex]
Therefore, the value of [tex]\(\frac{x^{m+n} \times y^{n-m}}{x^{m-n} \times y^{n+m}}\)[/tex] is [tex]\(\boxed{1024}\)[/tex].
