Problem 56 - A Graphical Approach to College Algebra

Question

Find each product. Assume that all variables represent positive real numbers. $$-5 y\left(3 y^{9 / 10}+4 y^{3 / 10}\right)$$

Step-by-Step Solution

Answer

The product is \(-15y^{19/10} - 20y^{13/10}\).

Step 1: Distribute the Negative Sign

Start by distributing the negative sign from \(-5y\) to each term inside the parentheses using distribution property. This is done by multiplying \(-5y\) with \(3y^{9/10}\) and \(-5y\) with \(4y^{3/10}\).

Step 2: Multiply First Term

Multiply \(-5y\) by \(3y^{9/10}\). This results in \(-5 \cdot 3 \cdot y \cdot y^{9/10} = -15y^{1 + 9/10}\). Because \(y\) has an implied power of \(1\), add the exponents: \(-15y^{10/10 + 9/10} = -15y^{19/10}\).

Step 3: Multiply Second Term

Multiply \(-5y\) by \(4y^{3/10}\). This results in\(-5 \cdot 4 \cdot y \cdot y^{3/10} = -20 y^{1 + 3/10}\). Add the exponents for \(y\): \(-20y^{10/10 + 3/10} = -20y^{13/10}\).

Step 4: Combine the Products

Now, combine the two products obtained from the previous steps: \(-15y^{19/10} - 20y^{13/10}\). This is the expression representing the product of the original problem contents.

Key Concepts

Polynomial multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another. In our exercise, the polynomial multiplication is essentially the process of distributing the term \(-5y\) across the terms inside the parenthesis. This means we take \(-5y\) and multiply it with each term separately, which are \(3y^{9/10}\) and \(4y^{3/10}\).
When performing polynomial multiplication, it's crucial to handle both the coefficients (the numbers) and the variables correctly. Let's break this down:

• Multiply the coefficients: Here, we multiply \(-5\) with \(3\) and \(-5\) with \(4\).
• Apply the exponent rules to manage the variable parts, combining powers of\( y \) when multiplying.

Grasping this concept enables us to solve more complex expressions by breaking them down into simpler operations using these multiplication fundamentals.

Exponent rules

Exponent rules, also known as laws of exponents, are used for simplifying expressions that involve powers of the same base. Commonly, you’ll encounter situations where variables have exponents, much like in our exercise.
The key exponent rule applied here is: when multiplying two powers that have the same base, you add the exponents together.
For example:

• Given \(y\) has an implicit exponent of 1, multiplying \(y\) with \(y^{9/10}\), we get \(y^{1 + 9/10} = y^{19/10}\).
• Similarly, for \(4y^{3/10}\), multiplying with \(y\), yields \(y^{1 + 3/10} = y^{13/10}\).

Understanding and applying these exponent rules allows you to elegantly handle more complex mathematical expressions by dealing precisely with the powers.

Distributive property

The distributive property is a powerful mathematical tool that facilitates carrying out multiplication across added or subtracted terms within parentheses.
This property is expressed as \(a(b + c) = ab + ac\). Essentially, it involves spreading out a single term across multiple terms inside a bracket through multiplication.
In context of the given exercise:

• We see its application when \(-5y\) is multiplied by each term inside the parenthesis, namely \(3y^{9/10}\) and \(4y^{3/10}\). The result is two distinct products, \(-15y^{19/10}\) and \(-20y^{13/10}\), which are then combined.

By understanding the distributive property, you are empowered to transfer any operation through terms inside parentheses, making it easier to manage and simplify expressions.