What Is $4 \sqrt{32} + 6 \sqrt{50}$ In Simplified Radical Form?Enter Your Answer In The Box.$\square$

What is $4 \sqrt{32} + 6 \sqrt{50}$ in simplified radical form?

The problem requires us to simplify the expression $4 \sqrt{32} + 6 \sqrt{50}$ and express it in simplified radical form. To do this, we need to first simplify the radicals individually and then combine them.

Simplifying Radicals

Simplifying $\sqrt{32}$

To simplify $\sqrt{32}$, we need to find the largest perfect square that divides 32. We know that $32 = 16 \times 2$, and since $16$ is a perfect square, we can write $\sqrt{32}$ as $\sqrt{16 \times 2}$.

Using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can simplify $\sqrt{32}$ as follows:

$\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4 \sqrt{2}$

Simplifying $\sqrt{50}$

To simplify $\sqrt{50}$, we need to find the largest perfect square that divides 50. We know that $50 = 25 \times 2$, and since $25$ is a perfect square, we can write $\sqrt{50}$ as $\sqrt{25 \times 2}$.

Using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can simplify $\sqrt{50}$ as follows:

$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}$

Combining the Simplified Radicals

Now that we have simplified the radicals individually, we can combine them to get the final expression.

$4 \sqrt{32} + 6 \sqrt{50} = 4 \sqrt{16 \times 2} + 6 \sqrt{25 \times 2}$

Using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can simplify the expression as follows:

$4 \sqrt{16 \times 2} + 6 \sqrt{25 \times 2} = 4 \times 4 \sqrt{2} + 6 \times 5 \sqrt{2}$

Simplifying further, we get:

$16 \sqrt{2} + 30 \sqrt{2} = 46 \sqrt{2}$

Conclusion

Therefore, the simplified radical form of the expression $4 \sqrt{32} + 6 \sqrt{50}$ is $46 \sqrt{2}$.

Key Takeaways

• To simplify radicals, we need to find the largest perfect square that divides the number inside the radical.
• We can use the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify radicals.
• Combining simplified radicals involves multiplying the coefficients and adding the radicals.

Practice Problems

1. Simplify the expression $\sqrt{72} + \sqrt{48}$.
2. Simplify the expression $3 \sqrt{27} + 2 \sqrt{36}$.
3. Simplify the expression $\sqrt{75} + \sqrt{12}$.

Answer Key

1. $\sqrt{72} + \sqrt{48} = 6 \sqrt{2} + 4 \sqrt{3}$
2. $3 \sqrt{27} + 2 \sqrt{36} = 9 \sqrt{3} + 12 \sqrt{4} = 9 \sqrt{3} + 24$
3. $\sqrt{75} + \sqrt{12} = 5 \sqrt{3} + 2 \sqrt{3} = 7 \sqrt{3}$
   Q&A: Simplifying Radicals ==========================

Frequently Asked Questions

Q: What is the largest perfect square that divides 32?

A: The largest perfect square that divides 32 is 16.

Q: How do I simplify $\sqrt{32}$?

A: To simplify $\sqrt{32}$, we can write it as $\sqrt{16 \times 2}$ and then simplify further using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This gives us $\sqrt{32} = \sqrt{16} \times \sqrt{2} = 4 \sqrt{2}$.

Q: What is the largest perfect square that divides 50?

A: The largest perfect square that divides 50 is 25.

Q: How do I simplify $\sqrt{50}$?

A: To simplify $\sqrt{50}$, we can write it as $\sqrt{25 \times 2}$ and then simplify further using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This gives us $\sqrt{50} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}$.

Q: How do I combine simplified radicals?

A: To combine simplified radicals, we need to multiply the coefficients and add the radicals. For example, if we have $4 \sqrt{32} + 6 \sqrt{50}$, we can simplify the radicals individually and then combine them to get $46 \sqrt{2}$.

Q: What is the simplified radical form of the expression $4 \sqrt{32} + 6 \sqrt{50}$?

A: The simplified radical form of the expression $4 \sqrt{32} + 6 \sqrt{50}$ is $46 \sqrt{2}$.

Q: How do I simplify the expression $\sqrt{72} + \sqrt{48}$?

A: To simplify the expression $\sqrt{72} + \sqrt{48}$, we need to find the largest perfect square that divides each number inside the radical. We can then simplify further using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This gives us $\sqrt{72} + \sqrt{48} = 6 \sqrt{2} + 4 \sqrt{3}$.

Q: How do I simplify the expression $3 \sqrt{27} + 2 \sqrt{36}$?

A: To simplify the expression $3 \sqrt{27} + 2 \sqrt{36}$, we need to find the largest perfect square that divides each number inside the radical. We can then simplify further using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This gives us $3 \sqrt{27} + 2 \sqrt{36} = 9 \sqrt{3} + 12 \sqrt{4} = 9 \sqrt{3} + 24$.

Q: How do I simplify the expression $\sqrt{75} + \sqrt{12}$?

A: To simplify the expression $\sqrt{75} + \sqrt{12}$, we need to find the largest perfect square that divides each number inside the radical. We can then simplify further using the property of radicals that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. This gives us $\sqrt{75} + \sqrt{12} = 5 \sqrt{3} + 2 \sqrt{3} = 7 \sqrt{3}$.

Conclusion

Simplifying radicals is an important concept in mathematics that can be used to simplify expressions and solve equations. By understanding the properties of radicals and how to simplify them, we can solve a wide range of problems in mathematics.