Problem 57 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-56}}{\sqrt{-7}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 2\sqrt{2} \).

Step 1: Express each square root in terms of i

The expression is given as \( \frac{\sqrt{-56}}{\sqrt{-7}} \). Rewrite each square root using \( i \) as follows:\[ \sqrt{-56} = i \sqrt{56} \] \[ \sqrt{-7} = i \sqrt{7} \]

Step 2: Substitute i expressions into the fraction

Substitute the expressions from Step 1 into the original fraction:\[ \frac{i \sqrt{56}}{i \sqrt{7}} \]

Step 3: Simplify the expression

Cancel out \( i \) from the numerator and the denominator:\[ \frac{\sqrt{56}}{\sqrt{7}} \] The expression simplifies to \( \sqrt{\frac{56}{7}} \).

Step 4: Simplify the square roots

Compute the division inside the square root:\[ \sqrt{\frac{56}{7}} = \sqrt{8} \]

Step 5: Simplify \(\sqrt{8}\) further

Simplify \( \sqrt{8} \) as follows:\[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \]

Step 6: Final Result

The simplified expression is:\[ 2\sqrt{2} \]

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is an essential concept in complex numbers. It is defined such that \( i^2 = -1 \). This definition allows us to work with the square roots of negative numbers, which are not possible within the system of real numbers.

• Expressing Square Roots of Negative Numbers: To express the square root of a negative number, we use \( i \). For instance, \( \sqrt{-a} \) can be rewritten as \( i\sqrt{a} \).
• Usage in Calculations: It enables us to simplify expressions involving negative square roots and perform operations like addition, subtraction, and multiplication in the realm of complex numbers.
• Multiplication Rules: When multiplying by \( i \), remember that \( i^2 = -1 \), which affects the overall sign and value of the expression.

Without the imaginary unit, mathematical expressions involving \( \sqrt{-1} \) would remain undefined in most real-world applications.

Simplifying Expressions

Simplifying expressions, especially those involving complex numbers, requires a step-by-step approach to reduce them to their simplest form. This involves a few core strategies.

• Cancellation: When you have the same factor in both the numerator and denominator of a fraction, it can be canceled out, simplifying the expression considerably. For instance, in the given problem, \( i \) is present in both, leading us to remove it from the fraction altogether.
• Combining Like Terms: If there are similar radical terms, they can be combined just like regular algebraic terms to streamline the expression further.
• Factorization: The problem often involves identifying factors within radicals, which leads to breaking them up into more manageable parts – a critical step in reaching a simplified form, as evidenced by turning \( \sqrt{8} \) into \( 2\sqrt{2} \).

This structured approach ensures that the final expression is both accurate and understandable.

Radical Expressions

Radical expressions involve the root of a number, such as square roots or cube roots. These expressions can initially look complicated, but with systematic simplification, they become easier to interpret and solve.

• Breaking Down Radicals: One of the primary steps is to express radicals in terms of their prime factors. For example, \( \sqrt{8} \) can be broken into \( \sqrt{4\cdot2} \), and since \( \sqrt{4} = 2 \), it simplifies to \( 2\sqrt{2} \).
• Rationalizing the Denominator: Although not present in this problem, often radicals in the denominator are simplified by multiplying by an equivalent radical to make the denominator rational.
• Square Root Properties: Understanding and applying properties of square roots, such as \( \sqrt{a\cdot b} = \sqrt{a}\cdot\sqrt{b} \), assists in the process of radical simplification.

These techniques make it feasible to handle complex radical expressions more efficiently, yielding simpler and more usable mathematical expressions.