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

Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to their most basic form. This involves breaking down the radicals into their prime factors and then simplifying the expression. In this case, we're given the expression $4 \sqrt{32}+6 \sqrt{50}$ and we need to simplify it to its radical form.

Simplifying the Radicals

To simplify the radicals, we need to find the prime factors of the numbers inside the square roots. Let's start by simplifying $\sqrt{32}$ and $\sqrt{50}$.

Simplifying $\sqrt{32}$

$\sqrt{32}$ can be simplified by finding the prime factors of 32. We can write 32 as $2^5$. Since we have a power of 2, we can take the square root of $2^4$ outside of the radical, leaving $2\sqrt{2}$ inside the radical.

\sqrt{32} = \sqrt{2^5} = \sqrt{2^4 \cdot 2} = 2\sqrt{2}

Simplifying $\sqrt{50}$

$\sqrt{50}$ can be simplified by finding the prime factors of 50. We can write 50 as $2 \cdot 5^2$. Since we have a power of 5, we can take the square root of $5^2$ outside of the radical, leaving $5\sqrt{2}$ inside the radical.

\sqrt{50} = \sqrt{2 \cdot 5^2} = 5\sqrt{2}

Simplifying the Expression

Now that we have simplified the radicals, we can substitute the simplified forms back into the original expression.

$$4 \sqrt{32}+6 \sqrt{50} = 4(2\sqrt{2}) + 6(5\sqrt{2})$$

We can now distribute the numbers outside the radicals to the terms inside the radicals.

$$4(2\sqrt{2}) + 6(5\sqrt{2}) = 8\sqrt{2} + 30\sqrt{2}$$

We can combine like terms by adding the coefficients of the like terms.

$$8\sqrt{2} + 30\sqrt{2} = 38\sqrt{2}$$

Conclusion

In conclusion, the simplified form of the expression $4 \sqrt{32}+6 \sqrt{50}$ is $38\sqrt{2}$. This is the most basic form of the expression, and it cannot be simplified further.

Final Answer

The final answer is: $\boxed{38\sqrt{2}}$

Understanding the Basics of Simplifying Radicals

Simplifying radicals and expressions is a crucial concept in mathematics, particularly in algebra and geometry. In the previous article, we explored how to simplify the expression $4 \sqrt{32}+6 \sqrt{50}$ to its most basic form. In this article, we'll delve deeper into the world of simplifying radicals and expressions, and answer some frequently asked questions.

Q&A: Simplifying Radicals and Expressions

Q: What is the difference between a radical and an expression?

A: A radical is a mathematical expression that represents the square root of a number. An expression, on the other hand, is a combination of numbers, variables, and mathematical operations. For example, $\sqrt{16}$ is a radical, while $2x + 3$ is an expression.

Q: How do I simplify a radical?

A: To simplify a radical, you need to find the prime factors of the number inside the radical. If the number can be expressed as a product of perfect squares, you can take the square root of the perfect squares outside of the radical, leaving the remaining factors inside the radical.

Q: What is the rule for simplifying radicals?

A: The rule for simplifying radicals is to find the prime factors of the number inside the radical and take the square root of the perfect squares outside of the radical. For example, $\sqrt{18}$ can be simplified as $\sqrt{9 \cdot 2}$, which becomes $3\sqrt{2}$.

Q: Can I simplify an expression with multiple radicals?

A: Yes, you can simplify an expression with multiple radicals by simplifying each radical individually and then combining the simplified radicals. For example, $4\sqrt{32} + 6\sqrt{50}$ can be simplified as $4(2\sqrt{2}) + 6(5\sqrt{2})$, which becomes $8\sqrt{2} + 30\sqrt{2}$, and finally $38\sqrt{2}$.

Q: How do I know if a radical can be simplified?

A: A radical can be simplified if the number inside the radical can be expressed as a product of perfect squares. If the number cannot be expressed as a product of perfect squares, the radical cannot be simplified.

Q: Can I simplify a radical with a variable?

A: Yes, you can simplify a radical with a variable by following the same rules as simplifying a radical with a number. For example, $\sqrt{16x}$ can be simplified as $\sqrt{16} \cdot \sqrt{x}$, which becomes $4\sqrt{x}$.

Common Mistakes to Avoid

When simplifying radicals and expressions, there are several common mistakes to avoid:

• Not finding the prime factors of the number inside the radical
• Not taking the square root of the perfect squares outside of the radical
• Not combining like terms
• Not following the order of operations

Conclusion

Simplifying radicals and expressions is a crucial concept in mathematics, and it requires a deep understanding of the rules and procedures involved. By following the rules and procedures outlined in this article, you can simplify radicals and expressions with ease and confidence.

Final Tips

• Always find the prime factors of the number inside the radical
• Always take the square root of the perfect squares outside of the radical
• Always combine like terms
• Always follow the order of operations

By following these tips, you can become a master of simplifying radicals and expressions and tackle even the most complex mathematical problems with ease.