To solve the expression [tex]\(\frac{x^{2n-3} \times (x^2)^{n+1}}{(x^4)^{-3}}\)[/tex] step-by-step, we will simplify each part of the expression one at a time and combine the results.
1. Simplify the numerator:
The numerator is [tex]\(x^{2n-3} \times (x^2)^{n+1}\)[/tex].
- For [tex]\((x^2)^{n+1}\)[/tex], we can use the power of a power rule, which states [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(x^2)^{n+1} = x^{2(n+1)} = x^{2n + 2}
\][/tex]
- Therefore, the numerator simplifies to:
[tex]\[
x^{2n-3} \times x^{2n+2}
\][/tex]
- Next, we use the product of powers rule, which states [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
x^{2n-3} \times x^{2n+2} = x^{(2n-3) + (2n+2)} = x^{4n - 1}
\][/tex]
2. Simplify the denominator:
The denominator is [tex]\((x^4)^{-3}\)[/tex].
- Using the power of a power rule again:
[tex]\[
(x^4)^{-3} = x^{4 \times (-3)} = x^{-12}
\][/tex]
3. Combine the simplified numerator and denominator:
The expression now is:
[tex]\[
\frac{x^{4n-1}}{x^{-12}}
\][/tex]
- We use the quotient of powers rule, which states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{x^{4n-1}}{x^{-12}} = x^{(4n-1) - (-12)} = x^{4n-1 + 12} = x^{4n + 11}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
x^{4n + 11}
\][/tex]
