Problem 10 - Algebra A Combined Function

Question

Multiply. \(\left(-\frac{3}{4} y^{7}\right)\left(\frac{1}{7} y^{4}\right)\)

Step-by-Step Solution

Answer

The product is \(-\frac{3}{28} y^{11}\).

Step 1: Multiply the Coefficients

To begin, we'll multiply the coefficients of the given expressions. Here, the coefficients are \(-\frac{3}{4}\) and \(\frac{1}{7}\). Multiply them as follows: \[\left(-\frac{3}{4}\right) \cdot \left(\frac{1}{7}\right) = -\frac{3}{28}.\]

Step 2: Multiply the Variables

Next, multiply the variable parts of the expressions, which are \(y^{7}\) and \(y^{4}\). According to the laws of exponents, when multiplying terms with the same base, add the exponents: \[(y^{7}) \cdot (y^{4}) = y^{7+4} = y^{11}.\]

Step 3: Combine the Results

Combine the results from Step 1 and Step 2 to form the final product: \[-\frac{3}{28} \cdot y^{11} = -\frac{3}{28} y^{11}.\]

Key Concepts

Multiplication of Fractions

Multiplying fractions might initially seem daunting, but it is quite straightforward. The key is to remember that fractions are multiplied directly across the top (numerators) and bottom (denominators). Let's break it down. Given two fractions, such as \(-\frac{3}{4}\) and \(\frac{1}{7}\), you multiply the numerators together and the denominators together. This means:

• Numerators: \(-3 \times 1 = -3\)
• Denominators: \(4 \times 7 = 28\)

This results in a product of \(-\frac{3}{28}\). Always check to see if the fraction can be simplified. In this case, \(-\frac{3}{28}\) is already in its simplest form. Keeping this straightforward process in mind will make fraction multiplication easy every time!

Laws of Exponents

The laws of exponents streamline the process of handling expressions with exponents. One of the most useful rules is when multiplying like bases, you simply add the exponents. This is called the 'product of powers' property. In the given example, we have \(y^7\) and \(y^4\). Since they share the same base \(y\), we add the exponents:

• Exponent Addition: \(7 + 4 = 11\)

So, \(y^7 \times y^4 = y^{11}\).
Remember this general rule: \(a^m \times a^n = a^{m+n}\). It allows you to simplify expressions quickly and efficiently whenever you encounter multiplication of powers with the same base.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions and finding solutions. In algebra, 'like terms' have identical variable parts. For instance, if two terms share the same variable and exponent, such as \(y^{11}\), they are considered like terms and can be combined.
In the problem provided, after applying the laws of exponents, you ended up with \(-\frac{3}{28}y^{11}\). If you had any other terms that were \(y^{11}\), you would combine them by simply adding or subtracting the coefficients.
This ability to combine like terms helps reduce the complexity of expressions and makes finding further solutions or simplifying results much easier. Always be on the lookout for like terms in your expressions to ensure you’re simplifying correctly!