Problem 46 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{6 x^{-2} y^{2}}{x y^{-3}} \cdot \frac{\left(4 x^{2} y\right)^{-2}}{x y^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3y}{8x^{7}}\)

Step 1: Simplify the expression by applying the rules of exponents

Take the given expression \(\frac{6 x^{-2} y^{2}}{x y^{-3}} \cdot \frac{\left(4 x^{2} y\right)^{-2}}{x y^{2}} \). Now, apply the rule \(a^{-n} = \frac{1}{a^n} \) for the negative exponents to convert them into positive form. So it would become \(\frac{6}{x^{2}}\frac{y^{2}}{y^{-3}} \cdot \frac{\left(\frac{1}{16x^{4} y^{2}}\right)}{x y^{2}} = \frac{6}{x^{2}} \cdot y^{5} \cdot \frac{1}{16x^{5} y^{4}}\)

Step 2: Merge the like terms

Now, we will merge like terms to simplify the expression further. The like terms here are the x and y terms. So, merging the x and y terms, we get\(\frac{6}{x^{2}} \cdot y^{5} \cdot \frac{1}{16x^{5} y^{4}} = \frac{6}{16x^{7} y^{-1}}\)

Step 3: Simplify the expression further

Finally, the expression simplifies to:\(\frac{6}{16x^{7} y^{-1}} = \frac{3}{8x^{7} y^{-1}}\). Apply the rule \(a^{-n} = \frac{1}{a^n} \) to finally get \(\frac{3}{8x^{7} y^{-1}} = \frac{3}{8x^{7}} \cdot y = \frac{3y}{8x^{7}}\)

Key Concepts

Negative exponents made simple

Negative exponents can look a bit scary, but they are actually quite easy to understand once you get the hang of it. A negative exponent, like in the expression \( a^{-n} \), means you take the reciprocal of the base raised to the positive exponent. In simple terms, \( a^{-n} = \frac{1}{a^n} \).
Let's say you have \( x^{-3} \). You can rewrite this as \( \frac{1}{x^3} \). Easy, right? So, instead of multiplying something by itself, you are dividing 1 by that base, multiplied by itself c times.
This rule helps us convert any negative exponents into a fraction which doesn't have signs that confuse us, making expressions easier to work with and understand. Any time you spot a negative exponent, remember to flip it into the denominator as a positive exponent so it's easier to handle.

Simplifying expressions one step at a time

When you're working with expressions like \( \frac{6 x^{-2} y^{2}}{x y^{-3}} \cdot \frac{(4 x^{2} y)^{-2}}{x y^{2}} \), the goal is to make them as easy to work with as possible.

The first step is often to deal with any negative exponents. As we've discussed, this means converting them into the reciprocal. Once the expression is free of negative exponents, combine like terms. This means adding or subtracting the exponents of like bases.
For example, if you have \( x^3 \) and \( x^5 \), you can combine them into \( x^{3+5} = x^8 \).
Finally, it's important to cancel out any common terms. In this exercise, after applying all these steps, you'll have a much simpler expression to work with, which is easier to understand and calculate.

A sneak peek into exponent rules

Exponent rules are like the grammar of math; they help us remember how to handle expressions correctly. These rules tell us how to simplify and manipulate exponents in almost any situation.

One of the key rules is the product of powers: when you multiply same bases, you can simply add the exponents. For instance, \( x^a \cdot x^b = x^{a+b} \).

• Quotient Rule: When dividing, subtract the exponents. For example, \( \frac{x^a}{x^b} = x^{a-b} \).
• Power of a Power: With expressions like \((x^a)^b\), multiply the exponents, resulting in \(x^{a\cdot b}\).
• Negative Exponent Rule: As we’ve said, flip it! \(a^{-n} = \frac{1}{a^n}\).

These rules are the building blocks that help in making complex expressions easy, even when they include tricky terms like negative exponents. Understanding these rules will make simplifying expressions like a piece of cake.