Certainly! Let's solve the given equation step-by-step:
The given equation is:
[tex]\[
\frac{x^{2n-3} \times x^{2(n+1)}}{(x^4)^{-3}} = \frac{(x^2)^2}{(x^4)^{-3}}
\][/tex]
Step 1: Simplify the numerator on the left-hand side.
According to the law of exponents, when you multiply powers with the same base, you add the exponents:
[tex]\[
x^{2n-3} \times x^{2(n+1)} = x^{(2n-3) + 2(n+1)}
\][/tex]
Simplify the exponent:
[tex]\[
(2n-3) + 2(n+1) = 2n - 3 + 2n + 2 = 4n - 1
\][/tex]
So, the numerator becomes:
[tex]\[
x^{4n-1}
\][/tex]
Step 2: Simplify the denominator on the left-hand side.
Using the property of exponents, [tex]\((x^a)^b = x^{ab}\)[/tex], we have:
[tex]\[
(x^4)^{-3} = x^{4 \times (-3)} = x^{-12}
\][/tex]
So, the left-hand side becomes:
[tex]\[
\frac{x^{4n-1}}{x^{-12}}
\][/tex]
Using the law of exponents which states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], the expression simplifies to:
[tex]\[
x^{(4n-1) - (-12)} = x^{4n - 1 + 12} = x^{4n + 11}
\][/tex]
Step 3: Simplify the right-hand side.
First, simplify the numerator:
[tex]\[
(x^2)^2 = x^{2 \times 2} = x^4
\][/tex]
Again, the denominator was already simplified as:
[tex]\[
(x^4)^{-3} = x^{-12}
\][/tex]
So, the right-hand side becomes:
[tex]\[
\frac{x^4}{x^{-12}}
\][/tex]
Using the law of exponents:
[tex]\[
x^{4 - (-12)} = x^{4 + 12} = x^{16}
\][/tex]
Step 4: Equate the simplified expressions from both sides.
We now have:
[tex]\[
x^{4n + 11} = x^{16}
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
4n + 11 = 16
\][/tex]
Step 5: Solve for [tex]\(n\)[/tex].
Subtract 11 from both sides:
[tex]\[
4n = 5
\][/tex]
Divide both sides by 4:
[tex]\[
n = \frac{5}{4} = 1.25
\][/tex]
Therefore, the solution is:
[tex]\[
n = 1.25
\][/tex]
