Simplify the expression completely: $\frac{18 x^{10} y^{11}}{32 x^6 y^7}$

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06-11-2026
Mathematics

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Simplify the expression completely: $\frac{18 x^{10} y^{11}}{32 x^6 y^7}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T00:47:50-05:00

Simplify the numerical fraction $\frac{18}{32}$ by dividing both the numerator and denominator by their greatest common divisor, 2, to obtain $\frac{9}{16}$.
Use the quotient rule of exponents, $\frac{a^m}{a^n} = a^{m-n}$, to simplify the variable terms: $\frac{x^{10}}{x^6} = x^{4}$ and $\frac{y^{11}}{y^7} = y^{4}$.
Combine the simplified coefficient and variable terms to arrive at the final result.
The fully simplified expression is $\boxed{\frac{9}{16} x^4 y^4}$.

Explanation

Analyzing the Expression

The goal is to simplify the algebraic expression $\frac{18 x^{10} y^{11}}{32 x^6 y^7}$. This involves two main components: simplifying the numerical coefficients and simplifying the variables with exponents using the laws of exponents. We assume $x$ and $y$ are non-zero real numbers, as the expression is defined as a fraction.

Simplifying the Numerical Coefficient

To simplify the coefficient $\frac{18}{32}$, we identify the greatest common divisor of the numerator and the denominator. Both 18 and 32 are divisible by 2.

Divide 18 by 2: $18 \div 2 = 9$.
Divide 32 by 2: $32 \div 2 = 16$.

Thus, the numerical fraction simplifies to $\frac{9}{16}$.

Applying the Laws of Exponents

Now we apply the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$.

For the $x$ variables: $\frac{x^{10}}{x^6} = x^{10-6} = x^4$.
For the $y$ variables: $\frac{y^{11}}{y^7} = y^{11-7} = y^4$.

Combining these results with the simplified coefficient gives the final simplified expression.

Final Conclusion

Putting all parts together, the simplified expression is $\frac{9}{16} x^4 y^4$. This is the final reduced form of the original expression. Therefore, the result is $\boxed{\frac{9}{16} x^4 y^4}$.

Examples

In real-world applications, such as physics or engineering formulas, complex expressions often arise from derived quantities. For example, if you are calculating the ratio of two physical power outputs modeled by monomials like $18 x^{10} y^{11}$ and $32 x^6 y^7$, simplifying the expression allows for much easier computation and data interpretation. By reducing the complexity of the formula to $\frac{9}{16} x^4 y^4$, scientists can quickly estimate the relationship between variables and understand the system behavior more efficiently.