Question

Multiply and simplify. $$5 i(2+3 i)$$

Short answer

-15 + 10i

Step-by-step solution

Apply Distributive Property

Multiply the imaginary number 5i by each term inside the parenthesis (2+3i).

Multiply the Real Part

Multiply 5i by 2 to get 10i, the real part of the product.

Multiply the Imaginary Part

Multiply 5i by 3i to get 15i^2. Since i^2 equals -1, this results in -15.

Combine the Results

Combine the real part and the imaginary part of the product to get the simplified expression.

Key concepts

The Distributive Property in Complex Numbers

Understanding the distributive property is essential when multiplying complex numbers, as it allows us to multiply a single term by each term in a binomial effectively. In the provided exercise, we applied this property to multiply the imaginary number 5i by each term inside the parenthesis, which included both a real part (2) and an imaginary part (3i).

With the distributive property, we see that multiplication spreads out or distributes across each term. This is why we perform two separate multiplications: 5i times 2, and 5i times 3i. Think of it as sharing equally; every term in the parenthesis gets an equal share of the multiplication with 5i. It's important to follow this rule carefully, as it ensures the accuracy of calculations involving complex numbers.

The distributive property is commonly represented as: \(a(b + c) = ab + ac\). It's a fundamental principle not just in complex numbers but across various fields of mathematics.

Understanding Imaginary Numbers

Imaginary numbers might seem a bit mystical, but they serve a practical purpose in mathematics. The standard form of an imaginary number is 'bi' where 'b' is a real number, and 'i' is the imaginary unit. We define 'i' as the square root of -1, which is not a value that can be found on the number line of real numbers.

Whenever you see 'i' squared in calculations (such as our Step 3, which resulted in 15i^2), remember that by definition, \(i^2 = -1\). This detail is crucial when simplifying complex expressions as it converts what seems like an imaginary component into a real number. In our example, the multiplication of the imaginary parts (5i and 3i) resulted in 15i^2, which simplifies to -15, a real number.

Imaginary numbers, when combined with real numbers, form complex numbers, capable of representing a multitude of phenomena in physics, engineering, and beyond.

Simplifying Complex Expressions

Simplifying complex expressions may appear daunting at first, but it's simply about combining like terms and applying basic arithmetic, albeit with a couple of extra rules due to the presence of the imaginary unit 'i'. Once you've used the distributive property to multiply out the parts of the complex number, as shown in our exercise, you'll often have a mix of real components and imaginary components.

The real components are any terms without 'i', and the imaginary components have 'i'. After multiplication, you should combine like terms, convert any instances of i^2 to -1, and sum everything up neatly. In our exercise, after multiplying - which produced 10i and -15 - we combined these to get the simplified complex expression: -15 + 10i.

This expression captures the essence of the complex number: a real part (-15) and an imaginary part (10i). Always remember, the final form for any complex number is a combination of its real and imaginary parts, usually represented as a + bi.