Problem 76 - Algebra A Combined Function

Question

$$ \begin{array}{|l|c|c|c|c|} \hline \text { Monomials } & \text { Add } & \text { Subtract } & \text { Multiply } & \text { Divide } \\ \hline 6 x, 3 x & 6 x+3 x=9 x & 6 x-3 x=3 x & 6 x \cdot 3 x=18 x^{2} & \frac{6 x}{3 x}=2 \\ \hline-12 x^{2}, 2 x & -12 x^{2}+2 x ; \text { can't be simplified } & -12 x^{2}-2 x ; \text { can't be simplified } & -12 x^{2} \cdot 2 x=-24 x^{3} & \frac{-12 x^{2}}{2 x}=-6 x \\ \hline 4 y^{3}, 4 y^{7} & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The results are: addition - no simplification, subtraction - no simplification, multiplication - \(16y^{10}\), division - \(\frac{1}{y^4}\).

Step 1: Addition of Monomials

The monomials given are \(4y^3\) and \(4y^7\), which do not have like terms (since their exponents differ). Thus, they cannot be added to result in a simplified expression. The expression remains \(4y^3 + 4y^7\).

Step 2: Subtraction of Monomials

Subtracting the monomials \(4y^3\) and \(4y^7\) again involves unlike terms because their exponents are different. Hence, the expression is not simplified further and remains \(4y^3 - 4y^7\).

Step 3: Multiplication of Monomials

To multiply the monomials \(4y^3\) and \(4y^7\), multiply the coefficients and add the exponents. This gives: \[4y^3 \cdot 4y^7 = (4 \cdot 4)y^{3+7} = 16y^{10}.\]

Step 4: Division of Monomials

To divide the monomials \(4y^3\) and \(4y^7\), divide the coefficients and subtract the exponents: \[\frac{4y^3}{4y^7} = \frac{4}{4}y^{3-7} = y^{-4}.\] Write this as \(\frac{1}{y^4}\) for standard form.

Key Concepts

Addition of Monomials

When it comes to adding monomials, the key is to focus on like terms. Like terms in algebra are those terms that have identical variable parts, meaning both the variable and its exponent must match. This is crucial for addition because only like terms can be combined to form a single monomial.

For instance, if you have monomials like \(6x\) and \(3x\), the variable part \(x\) is the same, allowing us to add their coefficients: \(6x + 3x = 9x\).

However, if the terms are not like, such as \(4y^3\) and \(4y^7\), addition is not possible without simplifying, as their exponents differ. Thus, their sum is simply expressed as \(4y^3 + 4y^7\).

Here's a quick checklist for adding monomials:

• Check if the terms are like (same variable and exponent).
• If they are like terms, add the coefficients.
• If not, rewrite the expression without combining them.

Subtraction of Monomials

Subtraction of monomials follows the same concept as addition; identifying like terms is essential. You must ensure that both the variable and its exponent are identical in the monomials being subtracted.

For example, with \(6x\) and \(3x\), subtract the coefficients: \(6x - 3x = 3x\).

On the other hand, attempting to subtract unlike terms such as \(4y^3 - 4y^7\) results in no simplification. The expression remains as \(4y^3 - 4y^7\) because the exponents of \(y\) are different.

Here's a simple process for subtracting monomials:

• Check for like terms (same variable and same exponent).
• Subtract their coefficients if they are like terms.
• If not, keep the expression as is without any subtraction.

Multiplication of Monomials

Unlike addition and subtraction, when multiplying monomials, you multiply both the coefficients and the variables, while adding the exponents of the variables.

For example, multiplying \(4y^3\) and \(4y^7\) involves:

• Multiplying the coefficients: \(4 \times 4 = 16\).
• Adding the exponents of \(y\): \(3 + 7 = 10\); hence, the result is \(16y^{10}\).

This process is harmonious with the laws of exponents, which state that when you multiply powers with the same base, you add their exponents.

The steps are straightforward:

• Multiply the coefficients.
• Add the exponents of the like bases.
• Simplify the resulting monomial fully.

Division of Monomials

Dividing monomials involves dividing their coefficients and subtracting the exponents of the common variable. It's important to understand the application of the law of exponents which states subtract the exponents of like bases during division.

For instance, to divide \(4y^3\) by \(4y^7\):

• First, divide the coefficients: \(\frac{4}{4} = 1\).
• Then, subtract the exponents of \(y\): \(3 - 7 = -4\).

This results in \(y^{-4}\). To express it in standard form, write it as \(\frac{1}{y^4}\). This negative exponent rule is crucial when dividing monomials with a larger exponent in the denominator.

To divide monomials effectively:

• Divide their coefficients.
• Subtract the exponents of like terms.
• Simplify and, if needed, express using positive exponents by flipping the base.