Simplify:$7 \sqrt{10} \cdot 4 \sqrt{10}$

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Understanding the Problem

When dealing with expressions involving square roots, it's essential to understand the properties of radicals and how they interact with multiplication. In this case, we're given the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$, and we're asked to simplify it.

Properties of Radicals

Before we dive into the simplification process, let's review some key properties of radicals:

• The product of two square roots is equal to the square root of the product of the numbers inside the radicals. In other words, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
• The product of a number and a square root is equal to the square root of the product of the number and the number inside the radical. In other words, $a \cdot \sqrt{b} = \sqrt{a^2 \cdot b}$.

Simplifying the Expression

Using the properties of radicals, we can simplify the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$ as follows:

$$7 \sqrt{10} \cdot 4 \sqrt{10} = \sqrt{7^2 \cdot 10} \cdot \sqrt{4^2 \cdot 10}$$

Applying the Properties of Radicals

Now, let's apply the properties of radicals to simplify the expression further:

$$\sqrt{7^2 \cdot 10} \cdot \sqrt{4^2 \cdot 10} = \sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10}$$

Simplifying the Radicals

Next, let's simplify the radicals by multiplying the numbers inside the radicals:

$$\sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10} = \sqrt{490} \cdot \sqrt{160}$$

Simplifying the Expression Further

Now, let's simplify the expression further by multiplying the numbers inside the radicals:

$$\sqrt{490} \cdot \sqrt{160} = \sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5}$$

Factoring the Numbers

Next, let's factor the numbers inside the radicals:

$$\sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5} = \sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)}$$

Simplifying the Radicals

Now, let's simplify the radicals by multiplying the numbers inside the radicals:

$$\sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)} = \sqrt{2^6 \cdot 7^2 \cdot 5^2}$$

Final Simplification

Finally, let's simplify the radicals by taking the square root of the numbers inside the radicals:

$$\sqrt{2^6 \cdot 7^2 \cdot 5^2} = 2^3 \cdot 7 \cdot 5 = 140 \cdot 7 = 980$$

Conclusion

In conclusion, the simplified expression is $980$. This is the final answer to the problem.

Additional Examples

Here are a few additional examples of simplifying expressions involving square roots:

• $3 \sqrt{2} \cdot 4 \sqrt{3} = \sqrt{3^2 \cdot 2} \cdot \sqrt{4^2 \cdot 3} = \sqrt{9 \cdot 2} \cdot \sqrt{16 \cdot 3} = \sqrt{18} \cdot \sqrt{48} = \sqrt{2 \cdot 3^2 \cdot 2^4 \cdot 3} = \sqrt{(2 \cdot 3^2 \cdot 2^4) \cdot 3} = \sqrt{2^5 \cdot 3^3} = 2^2 \cdot 3^2 = 36$
• $5 \sqrt{3} \cdot 2 \sqrt{5} = \sqrt{5^2 \cdot 3} \cdot \sqrt{2^2 \cdot 5} = \sqrt{25 \cdot 3} \cdot \sqrt{4 \cdot 5} = \sqrt{75} \cdot \sqrt{20} = \sqrt{3 \cdot 5^2 \cdot 2^2 \cdot 5} = \sqrt{(3 \cdot 5^2 \cdot 2^2) \cdot 5} = \sqrt{3 \cdot 5^3 \cdot 2^2} = \sqrt{3 \cdot 2^2 \cdot 5^3} = 2 \cdot 5^2 \cdot \sqrt{3 \cdot 5} = 50 \cdot \sqrt{15}$

Final Answer

The final answer is $\boxed{980}$.

$$

Understanding the Problem

When dealing with expressions involving square roots, it's essential to understand the properties of radicals and how they interact with multiplication. In this case, we're given the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$, and we're asked to simplify it.

Q&A

Q: What is the property of radicals that we use to simplify the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$?

A: The property of radicals that we use to simplify the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$ is the product of two square roots is equal to the square root of the product of the numbers inside the radicals. In other words, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: How do we simplify the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$ using the property of radicals?

A: We simplify the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$ by applying the property of radicals as follows:

$$7 \sqrt{10} \cdot 4 \sqrt{10} = \sqrt{7^2 \cdot 10} \cdot \sqrt{4^2 \cdot 10}$$

Q: What is the next step in simplifying the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$?

A: The next step in simplifying the expression $7 \sqrt{10} \cdot 4 \sqrt{10}$ is to multiply the numbers inside the radicals:

$$\sqrt{7^2 \cdot 10} \cdot \sqrt{4^2 \cdot 10} = \sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10}$$

Q: How do we simplify the expression $\sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10}$?

A: We simplify the expression $\sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10}$ by multiplying the numbers inside the radicals:

$$\sqrt{49 \cdot 10} \cdot \sqrt{16 \cdot 10} = \sqrt{490} \cdot \sqrt{160}$$

Q: What is the next step in simplifying the expression $\sqrt{490} \cdot \sqrt{160}$?

A: The next step in simplifying the expression $\sqrt{490} \cdot \sqrt{160}$ is to factor the numbers inside the radicals:

$$\sqrt{490} \cdot \sqrt{160} = \sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5}$$

Q: How do we simplify the expression $\sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5}$?

A: We simplify the expression $\sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5}$ by factoring the numbers inside the radicals:

$$\sqrt{2 \cdot 5 \cdot 7^2 \cdot 2^5 \cdot 5} = \sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)}$$

Q: What is the next step in simplifying the expression $\sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)}$?

A: The next step in simplifying the expression $\sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)}$ is to multiply the numbers inside the radicals:

$$\sqrt{(2 \cdot 7^2 \cdot 5) \cdot (2^5 \cdot 5)} = \sqrt{2^6 \cdot 7^2 \cdot 5^2}$$

Q: How do we simplify the expression $\sqrt{2^6 \cdot 7^2 \cdot 5^2}$?

A: We simplify the expression $\sqrt{2^6 \cdot 7^2 \cdot 5^2}$ by taking the square root of the numbers inside the radicals:

$$\sqrt{2^6 \cdot 7^2 \cdot 5^2} = 2^3 \cdot 7 \cdot 5 = 140 \cdot 7 = 980$$

Final Answer

The final answer is $\boxed{980}$.

Additional Examples

Here are a few additional examples of simplifying expressions involving square roots:

• $3 \sqrt{2} \cdot 4 \sqrt{3} = \sqrt{3^2 \cdot 2} \cdot \sqrt{4^2 \cdot 3} = \sqrt{9 \cdot 2} \cdot \sqrt{16 \cdot 3} = \sqrt{18} \cdot \sqrt{48} = \sqrt{2 \cdot 3^2 \cdot 2^4 \cdot 3} = \sqrt{(2 \cdot 3^2 \cdot 2^4) \cdot 3} = \sqrt{2^5 \cdot 3^3} = 2^2 \cdot 3^2 = 36$
• $5 \sqrt{3} \cdot 2 \sqrt{5} = \sqrt{5^2 \cdot 3} \cdot \sqrt{2^2 \cdot 5} = \sqrt{25 \cdot 3} \cdot \sqrt{4 \cdot 5} = \sqrt{75} \cdot \sqrt{20} = \sqrt{3 \cdot 5^2 \cdot 2^2 \cdot 5} = \sqrt{(3 \cdot 5^2 \cdot 2^2) \cdot 5} = \sqrt{3 \cdot 5^3 \cdot 2^2} = \sqrt{3 \cdot 2^2 \cdot 5^3} = 2 \cdot 5^2 \cdot \sqrt{3 \cdot 5} = 50 \cdot \sqrt{15}$

Final Answer

The final answer is $\boxed{980}$.