Problem 20 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \sqrt{-15} \cdot \sqrt{-5} $$

Step-by-Step Solution

Answer

The simplified form is \(-5\sqrt{3}\).

Step 1: Express Using Imaginary Unit

Recognize that the square root of a negative number involves the imaginary unit \(i\). Thus, express \(\sqrt{-15}\) as \(\sqrt{15} \cdot i\) and \(\sqrt{-5}\) as \(\sqrt{5} \cdot i\).

Step 2: Multiply the Expressions

Apply the associative and commutative properties of multiplication: \(\sqrt{-15} \cdot \sqrt{-5} = (\sqrt{15} \cdot i) \cdot (\sqrt{5} \cdot i)\). This simplifies to \((\sqrt{15} \cdot \sqrt{5}) \cdot (i \cdot i)\).

Step 3: Simplify Square Roots

Multiply \(\sqrt{15} \cdot \sqrt{5}\) to get \(\sqrt{75}\). Simplify \(\sqrt{75}\) as \(\sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}\).

Step 4: Simplify Imaginary Unit Multiplication

Recall that \(i \cdot i = i^2 = -1\). Thus, the expression becomes \(5\sqrt{3} \cdot (-1)\).

Step 5: Final Simplification

Multiply through by \(-1\) to arrive at the final simplified result: \(-5\sqrt{3}\).

Key Concepts

Understanding Complex Numbers

Complex numbers are fascinating entities in mathematics that extend our number system. They have the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).When dealing with square roots of negative numbers, we enter the realm of complex numbers. For instance, the square root of a negative number, such as \(\sqrt{-1}\), is represented as \(i\). Hence, \(\sqrt{-15}\) can be expressed as \(\sqrt{15} \, i\). This is a pivotal step in simplifying expressions that involve negative square roots.

• Real numbers reside on the number line, while complex numbers occupy a two-dimensional plane, with the real axis and the imaginary axis.
• Complex numbers allow us to solve equations that have no solutions in the realm of real numbers, such as \(x^2 + 1 = 0\).

This concept opens up an entire universe of mathematical possibilities, and using \(i\) helps us simplify puzzles like the original exercise.

Simplifying Radicals

Simplifying radicals is a crucial skill to make calculations easier and results more comprehensible. Radicals are expressions that involve roots, such as square roots, cube roots, etc. The key to simplifying radicals is to express them in terms of their prime factors.In our exercise, we need to simplify \(\sqrt{75}\). We break down 75 into its prime factors: \(75 = 25 \times 3 = 5^2 \times 3\). This allows us to rewrite \(\sqrt{75}\) as \(\sqrt{5^2 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}\). By pulling out the perfect square \(25\) from under the root, we arrive at a simpler expression.

• Prime factorization helps identify perfect squares that can be simplified outside the radical sign.
• The process of simplifying makes it easier to handle complex mathematical operations.

This method is a building block for algebra and even calculus, as it often appears in more advanced problem-solving and mathematical proofs.

Exploring Properties of Multiplication

The properties of multiplication are essential tools in simplifying and solving any mathematical problems. These properties include commutative, associative, distributive, identity, and inverse properties.

• Commutative Property: The order of factors does not affect the product, or \(a \cdot b = b \cdot a\).
• Associative Property: When multiplying three or more numbers, the way in which they are grouped does not change the product, i.e., \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity Property: Any number multiplied by 1 remains unchanged, so \(a \cdot 1 = a\).

In our exercise, we apply both associative and commutative properties to rearrange and group the elements in the expression \((\sqrt{15} \cdot i) \cdot (\sqrt{5} \cdot i)\) to simplify it effectively.These properties do wonders in simplifying our calculations and ensuring consistency in problem-solving. They also provide the foundation for more advanced mathematical concepts, proving indispensible as you progress in your studies.