Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$-6(5-3 i)$$

Step-by-Step Solution

Answer

The result of the operation \(-6(5-3 i)\) in standard form is \(-30 + 18i\).

Step 1: Distribute the multiplication

Distribute the \(-6\) to both the real and imaginary part by applying the distributive law (a * (b+c) = a * b + a * c). This means multiplying \(-6\) with both \(5\) and \(-3i\): \n \(-6 * 5 + -6 * -3i\)

Step 2: Compute the multiplication

Now, perform the multiplication: \n \(-30 + 18i\)

Step 3: Write the result in standard form

The standard form for a complex number is a + bi, where 'a' stands for the real part and 'bi' for the imaginary part. So providing the result in standard form, we have: \(-30 + 18i\)

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps simplify expressions. It involves distributing a single term across terms inside a parenthesis. In simple terms, if you have an equation like \( a(b+c) \), you distribute \( a \) to both \( b \) and \( c \). So, it becomes \( a \cdot b + a \cdot c \).

In context with complex numbers, let's see how it applies. Consider the expression \(-6(5-3i)\). To simplify, we apply the distributive property by multiplying \(-6\) with both the real number \(5\) and the imaginary part \(-3i\):

• Multiply \(-6\) by \(5\): \(-6 \times 5 = -30\).
• Multiply \(-6\) by \(-3i\): \(-6 \times -3i = 18i\).

By using this property, we've successfully distributed \(-6\) over both terms inside the parenthesis, simplifying the expression into two separate components.

Imaginary Numbers

Imaginary numbers are a fascinating aspect of mathematics, particularly when dealing with complex numbers. The imaginary unit is represented as \(i\) and is defined by the essential property that \(i^2 = -1\).

So, whenever you see a number multiplied by \(i\), you're dealing with an imaginary component.

• In the expression we simplified, the term \(-3i\) signifies that \(-3\) is associated with \(i\), making it imaginary.
• When we apply the distributive property, we multiply \(-3i\) with \(-6\), resulting in \(18i\).

This shows how the imaginary unit behaves under multiplication, transforming the values while still adhering to the rule \(i^2 = -1\). Remember, imaginary numbers are just as valid as real numbers in the realm of algebra. They open the door to a whole new dimension of mathematical calculation that can solve equations real numbers alone cannot.

Standard Form

In mathematics, complex numbers are typically presented in a format known as "standard form." This form is expressed as \(a + bi\), where \(a\) is the real component and \(bi\) represents the imaginary component.

Using standard form is beneficial because it clearly distinguishes between the real and imaginary parts of a number, making it easier to perform operations and understand results. For example, the result of the operation \(-6(5-3i)\) was simplified to \(-30 + 18i\).

• Here, \(-30\) is the real part, symbolized by \(a\).
• The \(18i\) is the imaginary part, noted as \(bi\).

This organized presentation ensures that any complex number is easy to read and interpret. It becomes straightforward to add, subtract, or visualize these numbers on a complex plane. Being comfortable with this form equips you with a foundation to tackle more challenging problems involving complex numbers.