Which Expression Is Equivalent To $\sqrt{-108} - \sqrt{-3}$?A. $5i \sqrt{3}$B. $6i \sqrt{3}$C. $7i \sqrt{3}$D. $8i \sqrt{3}$

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Understanding the Problem

The given problem involves simplifying an expression that contains square roots of negative numbers. To simplify this expression, we need to first understand the properties of square roots and how they can be manipulated. The expression $\sqrt{-108} - \sqrt{-3}$ can be simplified by first rewriting the square roots in terms of complex numbers.

Simplifying Square Roots of Negative Numbers

We know that the square root of a negative number can be expressed as a complex number in the form $ai$, where $a$ is a positive real number and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. Using this property, we can rewrite the square roots in the given expression as follows:

$\sqrt{-108} = \sqrt{-1 \cdot 108} = \sqrt{-1} \cdot \sqrt{108} = i \sqrt{108}$

$\sqrt{-3} = \sqrt{-1 \cdot 3} = \sqrt{-1} \cdot \sqrt{3} = i \sqrt{3}$

Substituting the Simplified Square Roots

Now that we have simplified the square roots, we can substitute these expressions back into the original expression:

$\sqrt{-108} - \sqrt{-3} = i \sqrt{108} - i \sqrt{3}$

Simplifying the Expression Further

To simplify the expression further, we can factor out the common term $i$:

$i \sqrt{108} - i \sqrt{3} = i (\sqrt{108} - \sqrt{3})$

Simplifying the Radicals

Now, we can simplify the radicals by factoring out the perfect square:

$\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}$

Substituting the Simplified Radical

Now that we have simplified the radical, we can substitute this expression back into the previous expression:

$i (\sqrt{108} - \sqrt{3}) = i (6 \sqrt{3} - \sqrt{3})$

Simplifying the Expression Further

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

$i (6 \sqrt{3} - \sqrt{3}) = i (5 \sqrt{3})$

Factoring Out the Common Term

Finally, we can factor out the common term $i$:

$i (5 \sqrt{3}) = 5i \sqrt{3}$

Conclusion

Therefore, the expression $\sqrt{-108} - \sqrt{-3}$ is equivalent to $5i \sqrt{3}$.

Answer

The correct answer is A. $5i \sqrt{3}$.

Discussion

This problem requires a good understanding of the properties of square roots and how they can be manipulated. It also requires the ability to simplify complex expressions and factor out common terms. The key to solving this problem is to recognize that the square roots of negative numbers can be expressed as complex numbers in the form $ai$, where $a$ is a positive real number and $i$ is the imaginary unit.

Related Problems

• Simplifying expressions with square roots of negative numbers
• Factoring out common terms in complex expressions
• Understanding the properties of square roots and complex numbers

Key Concepts

• Square roots of negative numbers
• Complex numbers
• Factoring out common terms
• Simplifying expressions with square roots

Practice Problems

• Simplify the expression $\sqrt{-16} - \sqrt{-9}$
• Simplify the expression $\sqrt{-25} - \sqrt{-4}$
• Simplify the expression $\sqrt{-36} - \sqrt{-9}$
  

Frequently Asked Questions

Q: What is the difference between a square root and a complex number?

A: A square root is a value that, when multiplied by itself, gives a specified value. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Q: How do I simplify an expression with a square root of a negative number?

A: To simplify an expression with a square root of a negative number, you can rewrite the square root in terms of a complex number. This involves multiplying the square root by the imaginary unit, $i$, to get a complex number in the form $ai$.

Q: What is the formula for simplifying a square root of a negative number?

A: The formula for simplifying a square root of a negative number is:

$\sqrt{-n} = i \sqrt{n}$

where $n$ is a positive integer.

Q: How do I simplify an expression with multiple square roots of negative numbers?

A: To simplify an expression with multiple square roots of negative numbers, you can apply the formula for simplifying a square root of a negative number to each square root individually. Then, you can combine the simplified expressions using the rules of arithmetic.

Q: What is the difference between $i$ and $-i$?

A: $i$ and $-i$ are both imaginary units, but they have opposite signs. When you multiply a complex number by $i$, you are essentially rotating the number by $90^\circ$ counterclockwise in the complex plane. When you multiply a complex number by $-i$, you are essentially rotating the number by $90^\circ$ clockwise in the complex plane.

Q: How do I simplify an expression with a square root of a negative number and a complex number?

A: To simplify an expression with a square root of a negative number and a complex number, you can apply the formula for simplifying a square root of a negative number to the square root, and then combine the simplified expression with the complex number using the rules of arithmetic.

Q: What is the formula for simplifying a complex number with a square root of a negative number?

A: The formula for simplifying a complex number with a square root of a negative number is:

$a + bi \sqrt{n} = (a + bi) \sqrt{n}$

where $a$ and $b$ are real numbers, and $n$ is a positive integer.

Q: How do I simplify an expression with multiple complex numbers and square roots of negative numbers?

A: To simplify an expression with multiple complex numbers and square roots of negative numbers, you can apply the formulas for simplifying complex numbers and square roots of negative numbers to each term individually. Then, you can combine the simplified expressions using the rules of arithmetic.

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

A: A real number is a value that can be expressed in the form $a$, where $a$ is a real number. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify an expression with a real number and a complex number?

A: To simplify an expression with a real number and a complex number, you can combine the two numbers using the rules of arithmetic.

Q: What is the formula for simplifying a real number and a complex number?

A: The formula for simplifying a real number and a complex number is:

$a + (b + ci) = (a + b) + (c)i$

where $a$ and $b$ are real numbers, and $c$ is a real number.

Q: How do I simplify an expression with multiple real numbers and complex numbers?

A: To simplify an expression with multiple real numbers and complex numbers, you can apply the formulas for simplifying real numbers and complex numbers to each term individually. Then, you can combine the simplified expressions using the rules of arithmetic.

Q: What is the difference between a rational number and a complex number?

A: A rational number is a value that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is non-zero. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify an expression with a rational number and a complex number?

A: To simplify an expression with a rational number and a complex number, you can combine the two numbers using the rules of arithmetic.

Q: What is the formula for simplifying a rational number and a complex number?

A: The formula for simplifying a rational number and a complex number is:

$\frac{a}{b} + (c + di) = \frac{a + bc + bdi}{b}$

where $a$ and $b$ are integers, and $c$ and $d$ are real numbers.

Q: How do I simplify an expression with multiple rational numbers and complex numbers?

A: To simplify an expression with multiple rational numbers and complex numbers, you can apply the formulas for simplifying rational numbers and complex numbers to each term individually. Then, you can combine the simplified expressions using the rules of arithmetic.

Q: What is the difference between a polynomial and a complex number?

A: A polynomial is a value that can be expressed in the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n, a_{n-1}, \ldots, a_1, a_0$ are real numbers and $x$ is a variable. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify an expression with a polynomial and a complex number?

A: To simplify an expression with a polynomial and a complex number, you can combine the two numbers using the rules of arithmetic.

Q: What is the formula for simplifying a polynomial and a complex number?

A: The formula for simplifying a polynomial and a complex number is:

$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 + (b + ci) = (a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0) + (b + ci)$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are real numbers, and $b$ and $c$ are real numbers.

Q: How do I simplify an expression with multiple polynomials and complex numbers?

A: To simplify an expression with multiple polynomials and complex numbers, you can apply the formulas for simplifying polynomials and complex numbers to each term individually. Then, you can combine the simplified expressions using the rules of arithmetic.

Q: What is the difference between a trigonometric function and a complex number?

A: A trigonometric function is a value that can be expressed in the form $\sin x$, $\cos x$, or $\tan x$, where $x$ is a real number. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify an expression with a trigonometric function and a complex number?

A: To simplify an expression with a trigonometric function and a complex number, you can combine the two numbers using the rules of arithmetic.

Q: What is the formula for simplifying a trigonometric function and a complex number?

A: The formula for simplifying a trigonometric function and a complex number is:

$\sin x + (b + ci) = \sin x + (b + ci)$

where $x$ is a real number, and $b$ and $c$ are real numbers.

Q: How do I simplify an expression with multiple trigonometric functions and complex numbers?

A: To simplify an expression with multiple trigonometric functions and complex numbers, you can apply the formulas for simplifying trigonometric functions and complex numbers to each term individually. Then, you can combine the simplified expressions using the rules of arithmetic.

Q: What is the difference between a logarithmic function and a complex number?

A: A logarithmic function is a value that can be expressed in the form $\log x$, where $x$ is a positive real number. A complex number is a value that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify an expression with a logarithmic function and a complex number?

A: To simplify an expression with a logarithmic function and a complex number, you can combine the two numbers using the rules of arithmetic.

Q: What is the formula for simplifying a logarithmic function and a complex number?

A: The formula for simplifying a logarithmic function and a complex number is:

$\log x + (b + ci) = \log x + (b + ci)$

where $x$ is