Problem 35 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \left(12 x^{2} y^{4}\right)\left(\frac{1}{2} x^{5} y\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(6x^7y^5\).

Step 1: Distribute the Coefficients

First, distribute the coefficients of the terms. We have \(12\) and \(\frac{1}{2}\). When you multiply these together: \(12 \times \frac{1}{2} = 6\).

Step 2: Simplify the Exponents of 'x'

Next, simplify the exponents of \(x\). Using the property of exponents \(a^m \cdot a^n = a^{m+n}\), combine \(x^2\) and \(x^5\):\[x^{2+5} = x^7\]

Step 3: Simplify the Exponents of 'y'

Now, simplify the exponents of \(y\). Again using the property of exponents:\[y^4 \cdot y^1 = y^{4+1} = y^5\]

Step 4: Write the Simplified Expression

Combine all parts. The simplified expression is given by multiplying the coefficient and the simplified exponents:\[6x^7y^5\]

Key Concepts

Simplifying Expressions

Simplifying expressions is like cleaning up a messy room: we're looking to arrange things neatly to make the expression easier to handle. When you're simplifying, you should

• Combine like terms
• Use algebraic rules

In the exercise, the expression \[\left(12 x^2 y^4\right)\left(\frac{1}{2} x^5 y\right)\]is simplified by first dealing with the numbers separately, then tidying up the exponents. By combining the coefficients 12 and \(\frac{1}{2}\), we get 6. This decrease in value exemplifies simplifying, where reduction to the simplest terms is key.

Multiplying Exponents

Understanding how to multiply exponents is fundamental when simplifying expressions like the one in the exercise. If you have the same base, the rules tell us to add the exponents instead of performing the multiplication one step at a time. For example, the expression \[x^2 \cdot x^5\]becomes \[x^{2+5} = x^7\].Similarly, when dealing with \[y^4 \cdot y^1\],we sum the exponents to get \[y^5\].Remember to focus on the bases being identical before applying this rule. It simplifies calculations and makes the overall expression cleaner.

Negative Exponents

Negative exponents can be tricky, but they simply mean the reciprocal of those with positive exponents. Suppose you encounter a term with a negative exponent, like \[x^{-n}\].This can be expressed as \[\frac{1}{x^n}\].This technique is vital during simplification since it eliminates negative exponents, making expressions more straightforward. In our specific exercise, there are no negative exponents to start with, so this rule isn't applied directly. However, by keeping this concept in mind, you can effectively manage expressions in any mathematical context.

Algebra

Algebra serves as the foundation for manipulating mathematical expressions. It involves using letters and symbols, such as \(x\) and \(y\), to represent numbers within equations. This flexibility allows for generalized problem solving and the ability to tackle a wide range of math problems. In our exercise, we leveraged simple algebraic principles to

• Combine like bases using exponent rules
• Multiply different terms

Both strategies showcase how algebra is more than mere number manipulation—it's a tool to systematically simplify and solve complex expressions.