Let's break down the problem and the reasoning behind it:

You want to solve 4√6 + 8√4 and also discuss whether the result is rational or irrational.

Step 1: Simplify the expression

1. 8√4: Since the square root of 4 is 2, we have: $8\sqrt{4} = 8 \times 2 = 16$

2. 4√6: This term remains as it is, since we cannot simplify √6 further.

Now we can rewrite the expression: $4\sqrt{6} + 16$

Step 2: Determine whether the result is rational or irrational

• 4√6 is irrational, because √6 is irrational (it cannot be expressed as a fraction of integers).
• 16 is rational, because it is a whole number.

When we add an irrational number (4√6) to a rational number (16), the result will always be irrational. This is because the sum of a rational number and an irrational number is always irrational.

Final Answer:

The answer 4√6 + 16 is irrational.

Now, let's evaluate the four provided options:

• The answer twelve square root of ten is rational because the sum of a rational number and irrational number is a rational number — Incorrect. The sum of a rational and irrational number is always irrational.

• The answer four square root of six plus sixteen is irrational because the sum of a rational number and irrational number is an irrational number — Correct. This is the correct explanation.

• The answer twelve square root of ten is irrational because the sum of a rational number and irrational number is an irrational number — Incorrect. This statement seems to be about an unrelated expression.

• The answer four square root of six plus sixteen is rational because the sum of a rational number and irrational number is a rational number — Incorrect. This contradicts the rule that the sum of a rational number and an irrational number is always irrational.

In conclusion:

The correct reasoning is that 4√6 + 16 is irrational, because the sum of a rational number (16) and an irrational number (4√6) is irrational.

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Would you like me to clarify anything or go into further detail?
Here are some related questions to expand your understanding:

1. What is the difference between rational and irrational numbers?
2. How can you tell if the sum of two numbers will be rational or irrational?
3. How do you simplify square roots of numbers like √6?
4. What happens when you multiply a rational number with an irrational number?
5. Can the sum of two irrational numbers ever be rational?

Tip: Always remember, the sum or product of a rational and irrational number is irrational, while the sum or product of two rational numbers is always rational.