Verified Answer :
- Expand the numerator \(\left(3x^2y^4\right)^3\) to get \(27x^6y^{12}\), then multiply by 4 to obtain \(108x^6y^{12}\).
- Expand the denominator \(\left(2x^3y^5\right)^4\) to get \(16x^{12}y^{20}\).
- Form the fraction \(\frac{108x^6y^{12}}{16x^{12}y^{20}}\) and simplify the constant \(\frac{108}{16}\) to \(\frac{27}{4}\).
- Use the quotient rule for exponents to reduce the variables, resulting in \(\boxed{\frac{27}{4x^6y^8}}\).
Explanation
- Analyzing the Expression
We are asked to simplify the expression \(\frac{4\left(3x^2y^4\right)^3}{\left(2x^3y^5\right)^4}\). To do this, we need to apply the laws of exponents and simplify both the numerator and the denominator step by step. Recall that for any base \(a\) and exponents \(n\) and \(m\):
- \((ab)^n = a^n b^n\)
- \((a^n)^m = a^{nm}\)
- \(\frac{a^n}{a^m} = a^{n-m}\)
We will expand the numerator and the denominator separately before dividing.
- Expanding the Numerator
First, we expand the numerator \(\left(3x^2y^4\right)^3\):
\[\left(3x^2y^4\right)^3 = 3^3 \cdot (x^2)^3 \cdot (y^4)^3 = 27 \cdot x^{2 \cdot 3} \cdot y^{4 \cdot 3} = 27x^6y^{12}\]
Next, we multiply this by the leading constant 4:
\[4 \cdot (27x^6y^{12}) = 108x^6y^{12}\]
So, the numerator is \(108x^6y^{12}\).
- Expanding the Denominator
Now, we expand the denominator \(\left(2x^3y^5\right)^4\):
\[\left(2x^3y^5\right)^4 = 2^4 \cdot (x^3)^4 \cdot (y^5)^4 = 16 \cdot x^{3 \cdot 4} \cdot y^{5 \cdot 4} = 16x^{12}y^{20}\]
So, the denominator is \(16x^{12}y^{20}\).
- Simplifying the Result
Now we combine the expanded numerator and denominator into a single fraction:
\[\frac{108x^6y^{12}}{16x^{12}y^{20}}\]
First, simplify the fraction \(\frac{108}{16}\) by dividing both terms by their greatest common divisor, which is 4:
\[\frac{108 \div 4}{16 \div 4} = \frac{27}{4}\]
Next, simplify the variables using the exponent rule \(\frac{a^n}{a^m} = a^{n-m}\):
For \(x\): \(\frac{x^6}{x^{12}} = x^{6-12} = x^{-6} = \frac{1}{x^6}\)
For \(y\): \(\frac{y^{12}}{y^{20}} = y^{12-20} = y^{-8} = \frac{1}{y^8}\)
Combining these, we get \(\frac{27}{4x^6y^8}\).
- Conclusion
Putting everything together, the simplified form of the original expression is \(\frac{27}{4x^6y^8}\). Thus, the final answer is \(\boxed{\frac{27}{4x^6y^8}}\).
Examples
In engineering and physics, algebraic expressions often model relationships between variables, such as calculating power efficiency or scaling factors in geometric models. Simplifying complex fractions like this one allows for easier computation and deeper insight into how changing one variable (like \(x\) or \(y\)) impacts the entire system. Mastery of these rules is foundational for working with formulas in fields ranging from structural engineering to data science, where efficient computation is key.