Which Expression Is Equivalent To $\frac{36 X^4 Y^5}{(3 X Y)^2}$?

**Which Expression is Equivalent to $\frac{36 x^4 y^5}{(3 x y)^2}$?**

Understanding the Problem

The given expression is a fraction with a numerator and a denominator. To simplify this expression, we need to apply the rules of exponents and simplify the fraction.

Breaking Down the Expression

The given expression is $\frac{36 x^4 y^5}{(3 x y)^2}$. We can start by simplifying the denominator using the rule of exponents that states $(ab)^n = a^n b^n$.

Simplifying the Denominator

Using the rule of exponents, we can rewrite the denominator as follows:

$(3 x y)^2 = 3^2 (x y)^2 = 9 x^2 y^2$

Now, the expression becomes $\frac{36 x^4 y^5}{9 x^2 y^2}$.

Simplifying the Fraction

To simplify the fraction, we can divide the numerator and the denominator by their greatest common factor (GCF). In this case, the GCF of 36 and 9 is 9.

$\frac{36 x^4 y^5}{9 x^2 y^2} = \frac{36 \div 9}{9 \div 9} \cdot \frac{x^4}{x^2} \cdot \frac{y^5}{y^2}$

Simplifying further, we get:

$4 x^{4-2} y^{5-2} = 4 x^2 y^3$

Conclusion

Therefore, the expression $\frac{36 x^4 y^5}{(3 x y)^2}$ is equivalent to $4 x^2 y^3$.

Frequently Asked Questions

Q: What is the rule of exponents that states $(ab)^n = a^n b^n$?

A: This rule states that when a product of two numbers is raised to a power, we can raise each number to that power separately.

Q: How do we simplify the denominator of the given expression?

A: We can simplify the denominator by applying the rule of exponents that states $(ab)^n = a^n b^n$.

Q: What is the greatest common factor (GCF) of 36 and 9?

A: The GCF of 36 and 9 is 9.

Q: How do we simplify the fraction $\frac{36 x^4 y^5}{9 x^2 y^2}$?

A: We can simplify the fraction by dividing the numerator and the denominator by their greatest common factor (GCF), which is 9.

Q: What is the final simplified expression?

A: The final simplified expression is $4 x^2 y^3$.

Q: What is the rule of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This rule states that when we multiply two numbers with the same base, we can add their exponents.

Q: How do we apply the rule of exponents $a^m \cdot a^n = a^{m+n}$ to the expression $x^4 \cdot x^2$?

A: We can apply the rule of exponents by adding the exponents, resulting in $x^{4+2} = x^6$.

Q: What is the final simplified expression for $x^4 \cdot x^2$?

A: The final simplified expression is $x^6$.

Q: How do we apply the rule of exponents $a^m \cdot a^n = a^{m+n}$ to the expression $y^5 \cdot y^2$?

A: We can apply the rule of exponents by adding the exponents, resulting in $y^{5+2} = y^7$.

Q: What is the final simplified expression for $y^5 \cdot y^2$?

A: The final simplified expression is $y^7$.

Q: How do we combine the simplified expressions $x^6$ and $y^7$?

A: We can combine the simplified expressions by multiplying them together, resulting in $x^6 y^7$.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression is $4 x^2 y^3$.

Additional Resources

• Rules of Exponents
• Simplifying Fractions
• Exponents and Powers

Conclusion

In conclusion, the expression $\frac{36 x^4 y^5}{(3 x y)^2}$ is equivalent to $4 x^2 y^3$. We simplified the expression by applying the rules of exponents and simplifying the fraction. We also answered frequently asked questions related to the problem and provided additional resources for further learning.