Problem 54 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-75} \sqrt{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(15i\).

Step 1: Express the Negative Square Root in Terms of i

To express the square root of a negative number in terms of the imaginary unit, we use the property that for any positive number \(a\), \(\sqrt{-a} = i\sqrt{a}\). Thus, \(\sqrt{-75} = i\sqrt{75}\).

Step 2: Simplify the Square Root of a Positive Number

Next, simplify \(\sqrt{75}\). Note that \(75 = 25 \times 3\), so \(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25}\sqrt{3}\). Since \(\sqrt{25} = 5\), we have \(\sqrt{75} = 5\sqrt{3}\).

Step 3: Combine Equivalent Terms

Substitute back to the expression obtained from Step 1: \(\sqrt{-75} = i \sqrt{75}\). Using the result from Step 2, we have \(\sqrt{-75} = i(5\sqrt{3}) = 5i\sqrt{3}\).

Step 4: Apply the Operation to Combine Expressions

Now, compute the expression \(\sqrt{-75} \sqrt{3}\) using the simplified form. Substitute \(\sqrt{-75} = 5i\sqrt{3}\) into the expression, which gives \(5i\sqrt{3} \cdot \sqrt{3}\).

Step 5: Simplify the Expression

Now, simplify \(5i\sqrt{3} \cdot \sqrt{3}\): since \(\sqrt{3} \cdot \sqrt{3} = 3\), we have \(5i \cdot 3 = 15i\). Thus, the expression simplifies to \(15i\).

Key Concepts

Imaginary Unit

In mathematics, an imaginary unit is a fundamental concept when dealing with complex numbers. The imaginary unit is typically represented by the symbol \(i\). Its main property is that \(i^2 = -1\). This might seem a bit strange at first, but it is incredibly useful for calculations involving square roots of negative numbers. These numbers don't have real solutions because no real number squared gives a negative result. Hence, mathematicians introduced \(i\) to help solve problems where negative square roots appear.

• Symbol for imaginary unit: \(i\)
• Key property: \(i^2 = -1\)
• Useful for calculations involving negative square roots

In the context of complex numbers, a number in the form \(a + bi\) is considered complex, where \(a\) is the real part and \(b\) is the imaginary part. This framework allows more flexibility in solving equations that involve negative square roots.

Simplifying Radicals

Simplifying radicals is another important step in manipulating complex numbers. A radical, often seen as a square root, can be simplified by breaking down the number into its prime factors. This simplification involves

• Finding factors of the number under the radical sign
• Identifying pairs of factors (for a square root)
• Taking one factor from each pair out of the radical

For example, consider \(\sqrt{75}\). This can be written as \(\sqrt{25 \times 3}\). Since \(\sqrt{25}\) is 5, the simplified version is \(5\sqrt{3}\). Simplifying radicals makes calculations easier and is especially helpful when combining radical terms.

Negative Square Root

Handling negative square roots can initially be confusing, but it becomes straightforward with the imaginary unit. When you encounter a negative number under a square root, you use the property \(\sqrt{-a} = i\sqrt{a}\), where \(a\) is positive. For instance, \(\sqrt{-75}\) becomes \(i\sqrt{75}\). This step relies on the imaginary unit to provide a meaningful answer rather than leaving an undefined expression.

• Negative under the square root: \(\sqrt{-a} = i\sqrt{a}\)
• Useful to convert to a comprehensible form

Once the negative is addressed with the imaginary unit, further simplification like that of \(\sqrt{75}\) can proceed as usual, leading to a manageable expression such as \(5i\sqrt{3}\). This manipulation shows that negative square roots can indeed be simplified and handled systematically within complex number operations.