Simplify The Expression To The Form $a + B I$:$-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32}$

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Understanding the Problem

The given expression involves the simplification of a combination of square roots, including both positive and negative values. The goal is to express the given expression in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as the square root of $-1$. To simplify the expression, we need to evaluate each square root individually and then combine the results.

Evaluating Square Roots

The first step is to evaluate each square root in the given expression.

Evaluating $\sqrt{144}$

$\sqrt{144}$ can be simplified as follows:

$\sqrt{144} = \sqrt{12^2} = 12$

Evaluating $\sqrt{-98}$

$\sqrt{-98}$ can be simplified as follows:

$\sqrt{-98} = \sqrt{-1 \times 98} = \sqrt{-1} \times \sqrt{98} = i\sqrt{98}$

Since $\sqrt{98} = \sqrt{2 \times 49} = \sqrt{2} \times \sqrt{49} = 7\sqrt{2}$, we can rewrite $\sqrt{-98}$ as:

$\sqrt{-98} = i \times 7\sqrt{2} = 7i\sqrt{2}$

Evaluating $\sqrt{81}$

$\sqrt{81}$ can be simplified as follows:

$\sqrt{81} = \sqrt{9^2} = 9$

Evaluating $\sqrt{-32}$

$\sqrt{-32}$ can be simplified as follows:

$\sqrt{-32} = \sqrt{-1 \times 32} = \sqrt{-1} \times \sqrt{32} = i\sqrt{32}$

Since $\sqrt{32} = \sqrt{2 \times 16} = \sqrt{2} \times \sqrt{16} = 4\sqrt{2}$, we can rewrite $\sqrt{-32}$ as:

$\sqrt{-32} = i \times 4\sqrt{2} = 4i\sqrt{2}$

Combining the Results

Now that we have evaluated each square root, we can combine the results to simplify the given expression.

$-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32} = -12 - 7i\sqrt{2} - 9 + 4i\sqrt{2}$

Simplifying the Expression

To simplify the expression further, we can combine like terms.

$-12 - 7i\sqrt{2} - 9 + 4i\sqrt{2} = -21 - 3i\sqrt{2}$

Conclusion

In this article, we simplified the given expression to the form $a + bi$. We evaluated each square root individually and then combined the results to obtain the final expression. The simplified expression is $-21 - 3i\sqrt{2}$.

Final Answer

The final answer is $\boxed{-21 - 3i\sqrt{2}}$.

Frequently Asked Questions

Q: What is the value of $\sqrt{144}$?

A: The value of $\sqrt{144}$ is $12$.

Q: What is the value of $\sqrt{-98}$?

A: The value of $\sqrt{-98}$ is $7i\sqrt{2}$.

Q: What is the value of $\sqrt{81}$?

A: The value of $\sqrt{81}$ is $9$.

Q: What is the value of $\sqrt{-32}$?

A: The value of $\sqrt{-32}$ is $4i\sqrt{2}$.

Q: How do you simplify the expression $-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32}$?

A: To simplify the expression, you need to evaluate each square root individually and then combine the results.

Step-by-Step Solution

Step 1: Evaluate $\sqrt{144}$

$\sqrt{144} = \sqrt{12^2} = 12$

Step 2: Evaluate $\sqrt{-98}$

$\sqrt{-98} = \sqrt{-1 \times 98} = \sqrt{-1} \times \sqrt{98} = i\sqrt{98}$

Since $\sqrt{98} = \sqrt{2 \times 49} = \sqrt{2} \times \sqrt{49} = 7\sqrt{2}$, we can rewrite $\sqrt{-98}$ as:

$\sqrt{-98} = i \times 7\sqrt{2} = 7i\sqrt{2}$

Step 3: Evaluate $\sqrt{81}$

$\sqrt{81} = \sqrt{9^2} = 9$

Step 4: Evaluate $\sqrt{-32}$

$\sqrt{-32} = \sqrt{-1 \times 32} = \sqrt{-1} \times \sqrt{32} = i\sqrt{32}$

Since $\sqrt{32} = \sqrt{2 \times 16} = \sqrt{2} \times \sqrt{16} = 4\sqrt{2}$, we can rewrite $\sqrt{-32}$ as:

$\sqrt{-32} = i \times 4\sqrt{2} = 4i\sqrt{2}$

Step 5: Combine the Results

$-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32} = -12 - 7i\sqrt{2} - 9 + 4i\sqrt{2}$

Step 6: Simplify the Expression

To simplify the expression further, we can combine like terms.

$-12 - 7i\sqrt{2} - 9 + 4i\sqrt{2} = -21 - 3i\sqrt{2}$

Related Topics

• Simplifying expressions with square roots
• Evaluating square roots of negative numbers
• Combining like terms in algebraic expressions

Key Takeaways

• To simplify an expression with square roots, you need to evaluate each square root individually.
• When evaluating square roots of negative numbers, you can rewrite them as the product of the square root of the absolute value and the imaginary unit.
• Combining like terms in algebraic expressions can help simplify the expression further.
  

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Frequently Asked Questions

Q: What is the value of $\sqrt{144}$?

A: The value of $\sqrt{144}$ is $12$.

Q: What is the value of $\sqrt{-98}$?

A: The value of $\sqrt{-98}$ is $7i\sqrt{2}$.

Q: What is the value of $\sqrt{81}$?

A: The value of $\sqrt{81}$ is $9$.

Q: What is the value of $\sqrt{-32}$?

A: The value of $\sqrt{-32}$ is $4i\sqrt{2}$.

Q: How do you simplify the expression $-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32}$?

A: To simplify the expression, you need to evaluate each square root individually and then combine the results.

Q: What is the final answer to the expression $-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32}$?

A: The final answer is $\boxed{-21 - 3i\sqrt{2}}$.

Q: Can you explain the steps to simplify the expression?

A: Yes, here are the steps:

1. Evaluate $\sqrt{144}$: $\sqrt{144} = \sqrt{12^2} = 12$
2. Evaluate $\sqrt{-98}$: $\sqrt{-98} = \sqrt{-1 \times 98} = \sqrt{-1} \times \sqrt{98} = i\sqrt{98}$
3. Evaluate $\sqrt{81}$: $\sqrt{81} = \sqrt{9^2} = 9$
4. Evaluate $\sqrt{-32}$: $\sqrt{-32} = \sqrt{-1 \times 32} = \sqrt{-1} \times \sqrt{32} = i\sqrt{32}$
5. Combine the results: $-\sqrt{144} - \sqrt{-98} - \sqrt{81} + \sqrt{-32} = -12 - 7i\sqrt{2} - 9 + 4i\sqrt{2}$
6. Simplify the expression: $-12 - 7i\sqrt{2} - 9 + 4i\sqrt{2} = -21 - 3i\sqrt{2}$

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary part, such as $3$, $-4$, or $5$. An imaginary number is a number that can be expressed with an imaginary part, such as $3i$, $-4i$, or $5i$.

Q: Can you explain the concept of the imaginary unit?

A: Yes, the imaginary unit is a mathematical concept that is used to extend the real numbers to the complex numbers. It is defined as the square root of $-1$, denoted by $i$. The imaginary unit is used to represent numbers that have an imaginary part.

Q: How do you add and subtract complex numbers?

A: To add and subtract complex numbers, you need to add and subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, you can add them as follows:

$(a + bi) + (c + di) = (a + c) + (b + d)i$

Similarly, you can subtract them as follows:

$(a + bi) - (c + di) = (a - c) + (b - d)i$

Q: Can you explain the concept of complex conjugates?

A: Yes, complex conjugates are pairs of complex numbers that have the same real part and the same imaginary part, but with opposite signs. For example, if you have a complex number $a + bi$, its complex conjugate is $a - bi$. Complex conjugates are used to simplify complex expressions and to find the square roots of complex numbers.

Related Topics

• Simplifying expressions with square roots
• Evaluating square roots of negative numbers
• Combining like terms in algebraic expressions
• Adding and subtracting complex numbers
• Complex conjugates

Key Takeaways

• To simplify an expression with square roots, you need to evaluate each square root individually.
• When evaluating square roots of negative numbers, you can rewrite them as the product of the square root of the absolute value and the imaginary unit.
• Combining like terms in algebraic expressions can help simplify the expression further.
• Complex conjugates are pairs of complex numbers that have the same real part and the same imaginary part, but with opposite signs.
• Adding and subtracting complex numbers involves adding and subtracting the real parts and the imaginary parts separately.