Problem 15 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$6-\sqrt{-98}$$

Step-by-Step Solution

Answer

The number expressed in terms of \(i\) is \(6 - i \cdot \sqrt{98}\).

Step 1: Identify the square root of a negative number

The given expression is \(6 - \sqrt{-98}\). We must pay attention to the square root of -98.

Step 2: Express the negative square root in terms of \(i\)

Remembering that the imaginary unit \(i\) is the square root of -1, we can say that \(\sqrt{-1} = i\). This means we can rewrite \(\sqrt{-98}\) as \(\sqrt{-1 \cdot 98}\) which can be simplified further as \(\sqrt{98} \cdot \sqrt{-1} = i \cdot \sqrt{98}\).

Step 3: Substitute back into the original expression

Substituting \(i \cdot \sqrt{98}\) for \(\sqrt{-98}\) in the expression, we get \(6 - i \cdot \sqrt{98}\).

Key Concepts

Complex Numbers

Complex numbers are a way to express numbers that include both real numbers and imaginary numbers. They are usually written in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here, \(i\) is the imaginary unit, representing \(\sqrt{-1}\).

• To form a complex number, just combine a real number with an imaginary number.
• In the problem, you express \(6 - \sqrt{-98}\) using \(i\), which makes the expression \(6 - i\sqrt{98}\).

Understanding complex numbers allows you to perform arithmetic easier on numbers that involve square roots of negative numbers. They are essential in many fields of science and engineering, as they offer a way to solve equations that don't have solutions within the set of real numbers.

Square Root

A square root asks for the number which, when multiplied by itself, will result in the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).

• When dealing with negative numbers, we cannot find a real number whose square is negative.
• This is where the imaginary unit \(i\) comes into play, as it represents \(\sqrt{-1}\).

In the original exercise, you need to take the square root of \(-98\). By using the property of square roots, \(\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\), you can rewrite \(\sqrt{-98}\) as \(\sqrt{98}\cdot\sqrt{-1}\) which simplifies to \(i\sqrt{98}\). Understanding square roots, especially with negative numbers, is a fundamental part of working with complex numbers.

Algebra

Algebra involves the use of symbols and numbers to express formulas and solve equations. It is a branch of mathematics that allows you to solve for unknown variables and explore different mathematical relationships.

• In solving the problem \(6 - \sqrt{-98}\), algebra helps by organizing and simplifying expressions.
• Substitution is a key technique used here to replace expressions with equivalent values like substituting \(i \sqrt{98}\) for \(\sqrt{-98}\).

Algebra is crucial for rearranging terms and simplifying complex numbers into their standard form. By tightening these skills, you can more effectively manage expressions involving complex numbers, which can often be challenging due to their imaginary components.