Simplify The Expression Below: 243 X 9 Y 15 \sqrt{243 X^9 Y^{15}} 243 X 9 Y 15 ​ $[ \begin{array}{ll} A) ; 3 X^4 Y^8 \sqrt{27 X} & \text{Orange} \ B) ; 3 X^3 Y^4 \sqrt{27} & \text{Purple} \ C) ; 9 X^3 Y^4 \sqrt{3} & \text{Light Green} \ D) ; 9 X^4 Y^8

Simplify the Expression: $\sqrt{243 x^9 y^{15}}$

Understanding the Problem

When simplifying an expression involving square roots, it's essential to identify perfect squares within the radicand. This allows us to simplify the expression by taking the square root of the perfect squares and leaving the remaining factors inside the square root. In this case, we're given the expression $\sqrt{243 x^9 y^{15}}$, and we need to simplify it.

Breaking Down the Radicand

To simplify the expression, we need to break down the radicand into its prime factors. The radicand is $243 x^9 y^{15}$. We can start by factoring $243$ into its prime factors: $243 = 3^5$. So, we can rewrite the radicand as $3^5 x^9 y^{15}$.

Identifying Perfect Squares

Now that we have the radicand broken down into its prime factors, we can identify the perfect squares. A perfect square is a number that can be expressed as the square of an integer. In this case, we have $3^5$, which is not a perfect square, but we can rewrite it as $3^4 \cdot 3$. The $3^4$ is a perfect square, and we can take its square root.

Simplifying the Expression

Now that we have identified the perfect square, we can simplify the expression. We can take the square root of the perfect square, which is $3^4$, and leave the remaining factors inside the square root. The remaining factors are $3 \cdot x^9 \cdot y^{15}$. We can simplify this further by taking the square root of $x^9$, which is $x^4 \cdot x$, and taking the square root of $y^{15}$, which is $y^8 \cdot y$.

Final Simplified Expression

After simplifying the expression, we get $\sqrt{243 x^9 y^{15}} = 3 x^4 y^8 \sqrt{3 \cdot x \cdot y}$. We can rewrite this as $3 x^4 y^8 \sqrt{3 x y}$.

Comparing with the Options

Now that we have simplified the expression, we can compare it with the options given. The correct option is:

• A) $3 x^4 y^8 \sqrt{27 x}$

This option matches our simplified expression, which is $3 x^4 y^8 \sqrt{3 x y}$. The only difference is that the option has $\sqrt{27 x}$ instead of $\sqrt{3 x y}$. However, we can rewrite $\sqrt{27 x}$ as $\sqrt{3 \cdot 3 \cdot 3 \cdot x}$, which is equal to $\sqrt{3^3 \cdot x}$. Since $\sqrt{3^3} = 3 \sqrt{3}$, we can rewrite $\sqrt{27 x}$ as $3 \sqrt{3} \cdot \sqrt{x}$. This is equal to $\sqrt{3 x y}$, which is the same as our simplified expression.

Conclusion

In conclusion, the correct option is A) $3 x^4 y^8 \sqrt{27 x}$. This option matches our simplified expression, and we can rewrite $\sqrt{27 x}$ as $\sqrt{3 x y}$, which is equal to our simplified expression.

Discussion

This problem requires a good understanding of perfect squares and how to simplify expressions involving square roots. It's essential to identify perfect squares within the radicand and take their square roots. The remaining factors should be left inside the square root. In this case, we had to break down the radicand into its prime factors, identify the perfect square, and simplify the expression.

Key Takeaways

• Identify perfect squares within the radicand.
• Take the square root of the perfect squares.
• Leave the remaining factors inside the square root.
• Break down the radicand into its prime factors.
• Simplify the expression by taking the square root of the perfect squares.

Practice Problems

Try simplifying the following expressions:

• $\sqrt{16 x^8 y^{12}}$
• $\sqrt{25 x^6 y^{10}}$
• $\sqrt{36 x^9 y^{18}}$

Answer Key

• $\sqrt{16 x^8 y^{12}} = 4 x^4 y^6 \sqrt{x y}$
• $\sqrt{25 x^6 y^{10}} = 5 x^3 y^5 \sqrt{x y}$
• $\sqrt{36 x^9 y^{18}} = 6 x^4 y^9 \sqrt{x y}$
  Simplify the Expression: $\sqrt{243 x^9 y^{15}}$ - Q&A

Q: What is the first step in simplifying the expression $\sqrt{243 x^9 y^{15}}$?

A: The first step in simplifying the expression is to break down the radicand into its prime factors. In this case, we can start by factoring $243$ into its prime factors: $243 = 3^5$. So, we can rewrite the radicand as $3^5 x^9 y^{15}$.

Q: How do we identify perfect squares within the radicand?

A: To identify perfect squares within the radicand, we need to look for numbers that can be expressed as the square of an integer. In this case, we have $3^5$, which is not a perfect square, but we can rewrite it as $3^4 \cdot 3$. The $3^4$ is a perfect square, and we can take its square root.

Q: What is the next step in simplifying the expression?

A: After identifying the perfect square, we can simplify the expression by taking the square root of the perfect square and leaving the remaining factors inside the square root. In this case, we can take the square root of $3^4$, which is $3^2$, and leave the remaining factors inside the square root.

Q: How do we simplify the remaining factors?

A: To simplify the remaining factors, we need to take the square root of each factor. In this case, we have $x^9$ and $y^{15}$. We can take the square root of $x^9$, which is $x^4 \cdot x$, and take the square root of $y^{15}$, which is $y^8 \cdot y$.

Q: What is the final simplified expression?

A: After simplifying the expression, we get $\sqrt{243 x^9 y^{15}} = 3 x^4 y^8 \sqrt{3 \cdot x \cdot y}$. We can rewrite this as $3 x^4 y^8 \sqrt{3 x y}$.

Q: How do we compare the simplified expression with the options given?

A: To compare the simplified expression with the options given, we need to look for the option that matches our simplified expression. In this case, the correct option is A) $3 x^4 y^8 \sqrt{27 x}$.

Q: What is the key takeaway from this problem?

A: The key takeaway from this problem is that we need to identify perfect squares within the radicand and take their square roots. We also need to leave the remaining factors inside the square root and simplify them by taking their square roots.

Q: What are some common mistakes to avoid when simplifying expressions involving square roots?

A: Some common mistakes to avoid when simplifying expressions involving square roots include:

• Not identifying perfect squares within the radicand.
• Not taking the square root of the perfect squares.
• Not leaving the remaining factors inside the square root.
• Not simplifying the remaining factors by taking their square roots.

Q: How can we practice simplifying expressions involving square roots?

A: We can practice simplifying expressions involving square roots by trying out different problems and exercises. We can also use online resources and practice tests to help us improve our skills.

Q: What are some real-world applications of simplifying expressions involving square roots?

A: Simplifying expressions involving square roots has many real-world applications, including:

• Calculating distances and lengths in geometry and trigonometry.
• Solving problems in physics and engineering.
• Working with algebraic expressions and equations.
• Solving problems in finance and economics.

Conclusion

In conclusion, simplifying expressions involving square roots requires a good understanding of perfect squares and how to simplify expressions. We need to identify perfect squares within the radicand, take their square roots, and leave the remaining factors inside the square root. We also need to simplify the remaining factors by taking their square roots. By practicing and mastering these skills, we can become proficient in simplifying expressions involving square roots and apply them to real-world problems.