Certainly! Let's break down and simplify the given expression step by step.
Given expression:
[tex]\[ \frac{\left(x^{19} y^{21}\right)^4}{\left(x^2 y^6\right)^2} \][/tex]
1. Simplify the numerator:
First, we need to raise both \(x^{19}\) and \(y^{21}\) to the power of 4:
[tex]\[ (x^{19} y^{21})^4 = (x^{19})^4 \cdot (y^{21})^4 \][/tex]
Use the power rule \((a^m)^n = a^{m \cdot n}\):
[tex]\[ x^{19 \cdot 4} \cdot y^{21 \cdot 4} \][/tex]
[tex]\[ x^{76} \cdot y^{84} \][/tex]
So the numerator simplifies to \(x^{76} \cdot y^{84}\).
2. Simplify the denominator:
Next, we raise both \(x^2\) and \(y^6\) to the power of 2:
[tex]\[ (x^2 y^6)^2 = (x^2)^2 \cdot (y^6)^2 \][/tex]
Using the power rule again:
[tex]\[ x^{2 \cdot 2} \cdot y^{6 \cdot 2} \][/tex]
[tex]\[ x^4 \cdot y^{12} \][/tex]
So the denominator simplifies to \(x^4 \cdot y^{12}\).
3. Combine the simplified numerator and denominator:
We now have:
[tex]\[ \frac{x^{76} y^{84}}{x^4 y^{12}} \][/tex]
We use the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \):
For \(x\):
[tex]\[ x^{76 - 4} = x^{72} \][/tex]
For \(y\):
[tex]\[ y^{84 - 12} = y^{72} \][/tex]
Therefore, the simplified expression is:
[tex]\[ x^{72} y^{72} \][/tex]
So, the final simplified expression is:
[tex]\[ x^{72} y^{72} \][/tex]
