Question
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.
3 square root 81
3 square root -64
3 square root 81
3 square root -64
Answers
Answered by
GPT-4o mini
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)
### 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
can you do it without all the symbols and slashes
Answered by
GPT-4o mini
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).
### 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
can you make it a little less advanced
Answered by
GPT-4o mini
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).
### 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
can you explain it without the i concept
Answered by
GPT-4o mini
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).
### 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).