Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work.

Question

3 square root 81
3 square root -64

Answer 1

Let's simplify each of the expressions step by step.

Expression 1: \( 3 \sqrt{81} \)

1. Find the square root: \[ \sqrt{81} = 9 \]

2. Multiply by 3: \[ 3 \cdot 9 = 27 \]

3. Conclusion: Thus, the simplified form of \( 3 \sqrt{81} \) is \( 27 \), which is a rational number.

Expression 2: \( 3 \sqrt{-64} \)

1. Recognize the negative inside the square root: Since there is a negative radicand, we incorporate the imaginary unit \( i \).

2. Find the square root: \[ \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i \]

3. Multiply by 3: \[ 3 \cdot 8i = 24i \]

4. Conclusion: Thus, the simplified form of \( 3 \sqrt{-64} \) is \( 24i \), which is an irrational number in the context of real numbers but can be classified as an imaginary number.

Final Answers:

• \( 3 \sqrt{81} = 27 \) (rational)
• \( 3 \sqrt{-64} = 24i \) (imaginary)

Answer 2

can you do it without all the symbols and slashes

Answer 3

Sure! Let's simplify each expression step by step without using symbols or slashes.

Expression 1: Three times the square root of eighty-one

1. First, we need to find the square root of eighty-one. The square root of eighty-one is nine.
2. Next, we multiply nine by three. Three times nine equals twenty-seven.
3. Therefore, the simplified form of three times the square root of eighty-one is twenty-seven, which is a rational number.

Expression 2: Three times the square root of negative sixty-four

1. Since the number inside the square root is negative, we will involve the imaginary unit, which is represented as "i."
2. The square root of negative sixty-four can be rewritten as the square root of sixty-four times the square root of negative one. The square root of sixty-four is eight, and the square root of negative one is "i."
3. Now we can combine these: the square root of negative sixty-four is eight times "i," which gives us eight "i."
4. We then multiply eight "i" by three. Three times eight "i" equals twenty-four "i."
5. Therefore, the simplified form of three times the square root of negative sixty-four is twenty-four "i," which is an imaginary number.

Final Answers:

• Three times the square root of eighty-one simplifies to twenty-seven (rational).
• Three times the square root of negative sixty-four simplifies to twenty-four "i" (imaginary).

Answer 4

can you make it a little less advanced

Answer 5

Of course! Let's break it down into simpler steps without using advanced terms.

Expression 1: Three times the square root of eighty-one

1. Find the square root of eighty-one: What number multiplied by itself gives you eighty-one? The answer is nine because nine times nine equals eighty-one.

2. Multiply by three: Now take that result (nine) and multiply it by three. So, three times nine equals twenty-seven.

3. Final answer: The answer for three times the square root of eighty-one is twenty-seven. This is a simple number, so we call it a rational number.

Expression 2: Three times the square root of negative sixty-four

1. Square root of negative sixty-four: Here, we have a negative number. The square root of negative sixty-four is a bit different. Since we can't take the square root of a negative number in regular math, we use a special concept called "i," which stands for an imaginary number.

2. Square root of sixty-four: The square root of sixty-four is eight because eight times eight equals sixty-four.

3. Combine the results: So, when we look for the square root of negative sixty-four, we get eight times "i." That’s written as eight "i."

4. Multiply by three: Now we take that result (eight "i") and multiply it by three. Three times eight "i" equals twenty-four "i."

5. Final answer: The answer for three times the square root of negative sixty-four is twenty-four "i." This means it’s an imaginary number, which is a little different from regular numbers.

Final Answers:

• Three times the square root of eighty-one is twenty-seven (a regular number).
• Three times the square root of negative sixty-four is twenty-four "i" (an imaginary number).

Answer 6

can you explain it without the i concept

Answer 7

Certainly! Let’s simplify each expression without using the imaginary unit "i" and without going into advanced concepts.

Expression 1: Three times the square root of eighty-one

1. Find the square root of eighty-one: The square root of eighty-one is the number that, when multiplied by itself, gives us eighty-one. That number is nine because nine times nine equals eighty-one.

2. Multiply by three: Now, we take nine and multiply it by three. Three times nine equals twenty-seven.

3. Final answer: So, three times the square root of eighty-one simplifies to twenty-seven. This is a regular number, which we call a rational number.

Expression 2: Three times the square root of negative sixty-four

1. Understanding negative numbers under the square root: When we see a square root with a negative number, like the square root of negative sixty-four, we can't find a real number that, when multiplied by itself, gives a negative result. In ordinary math, we say this doesn't have a solution among regular numbers.

2. Square root of sixty-four: If we just consider the positive part, the square root of sixty-four is eight because eight times eight equals sixty-four.

3. Result for negative square root: However, since we are dealing with negative sixty-four inside the square root, we can't find a regular number to use for the final expression. We typically say that the expression is not a real number since it can't be simplified like before.

4. Final answer: Therefore, three times the square root of negative sixty-four does not produce a regular number, and we conclude that this expression doesn't have a solution in the real number system.

Summary:

• Three times the square root of eighty-one simplifies to twenty-seven (which is a regular number).
• Three times the square root of negative sixty-four does not produce a solution in the regular number system (it is not a real number).