Simplify The Expression: $\[ -3 \sqrt{7r^3} \cdot 6 \sqrt{7r^2} \\]

Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier. In this case, we're given the expression $-3 \sqrt{7r^3} \cdot 6 \sqrt{7r^2}$, and we need to simplify it. To start, let's break down the expression and understand its components.

Breaking Down the Expression

The given expression consists of two square roots multiplied together. We can rewrite it as $-3 \cdot 6 \cdot \sqrt{7r^3} \cdot \sqrt{7r^2}$. This allows us to separate the coefficients and the square roots, making it easier to simplify.

Simplifying the Coefficients

The coefficients $-3$ and $6$ can be multiplied together to get $-18$. So, the expression becomes $-18 \cdot \sqrt{7r^3} \cdot \sqrt{7r^2}$.

Simplifying the Square Roots

Now, let's focus on simplifying the square roots. We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property to our expression, we get $-18 \cdot \sqrt{7r^3 \cdot 7r^2}$.

Combining Like Terms

The expression inside the square root can be simplified by combining like terms. We have $7r^3 \cdot 7r^2$, which can be rewritten as $49r^5$. So, the expression becomes $-18 \cdot \sqrt{49r^5}$.

Simplifying the Square Root

Now, we can simplify the square root by taking out the perfect square factor. We have $\sqrt{49r^5}$, which can be rewritten as $\sqrt{49} \cdot \sqrt{r^5}$. Since $\sqrt{49} = 7$, the expression becomes $-18 \cdot 7 \cdot \sqrt{r^5}$.

Final Simplification

Finally, we can simplify the expression by combining the coefficients and the square root. We have $-18 \cdot 7 \cdot \sqrt{r^5}$, which can be rewritten as $-126 \cdot \sqrt{r^5}$.

Conclusion

In conclusion, the simplified expression is $-126 \cdot \sqrt{r^5}$. This is the final answer to the given problem.

Key Takeaways

• When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier.
• We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify expressions.
• We can combine like terms inside the square root to simplify the expression.
• We can take out the perfect square factor from the square root to simplify the expression.

Final Answer

The final answer to the given problem is $-126 \cdot \sqrt{r^5}$.

Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier. In this case, we're given the expression $-3 \sqrt{7r^3} \cdot 6 \sqrt{7r^2}$, and we need to simplify it. To start, let's break down the expression and understand its components.

Q&A Session

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

A: The first step in simplifying the expression is to break it down into its components. We can rewrite the expression as $-3 \cdot 6 \cdot \sqrt{7r^3} \cdot \sqrt{7r^2}$, which allows us to separate the coefficients and the square roots.

Q: How do we simplify the coefficients?

A: We can multiply the coefficients $-3$ and $6$ together to get $-18$. So, the expression becomes $-18 \cdot \sqrt{7r^3} \cdot \sqrt{7r^2}$.

Q: What property of square roots can we use to simplify the expression?

A: We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property to our expression, we get $-18 \cdot \sqrt{7r^3 \cdot 7r^2}$.

Q: How do we simplify the expression inside the square root?

A: We can combine like terms inside the square root to simplify the expression. We have $7r^3 \cdot 7r^2$, which can be rewritten as $49r^5$. So, the expression becomes $-18 \cdot \sqrt{49r^5}$.

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

A: We can simplify the square root by taking out the perfect square factor. We have $\sqrt{49r^5}$, which can be rewritten as $\sqrt{49} \cdot \sqrt{r^5}$. Since $\sqrt{49} = 7$, the expression becomes $-18 \cdot 7 \cdot \sqrt{r^5}$.

Q: What is the final simplified expression?

A: Finally, we can simplify the expression by combining the coefficients and the square root. We have $-18 \cdot 7 \cdot \sqrt{r^5}$, which can be rewritten as $-126 \cdot \sqrt{r^5}$.

Key Takeaways

• When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier.
• We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify expressions.
• We can combine like terms inside the square root to simplify the expression.
• We can take out the perfect square factor from the square root to simplify the expression.

Final Answer

The final answer to the given problem is $-126 \cdot \sqrt{r^5}$.

Additional Resources

• For more information on simplifying expressions involving square roots, check out our article on [Simplifying Square Roots](link to article).
• For more practice problems on simplifying expressions, check out our [Math Practice Problems](link to practice problems).

Conclusion

In conclusion, simplifying expressions involving square roots requires a step-by-step approach. By breaking down the expression, using the property of square roots, combining like terms, and taking out the perfect square factor, we can simplify the expression and arrive at the final answer.