Problem 45 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \left(4 x^{2}\right)\left(8 x y^{3}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(4 x^{2}\right)\left(8 x y^{3}\right)\) using the product rule and the quotient rule of exponents. Answer: \(32x^3 y^{3}\)

Step 1: Recognize the product rule of exponents

The product rule of exponents states that for any non-zero base, when we multiply two terms with the same base, we add their exponents. So, for example, if we have \(a^m \cdot a^n\), then the product rule says the result will be \(a^{m+n}\). In our given expression, we have similar bases which are \(x\).

Step 2: Multiply the coefficients and apply the product rule

First, multiply the coefficients of the given expression, which are 4 and 8. The result is: $$ (4 \cdot 8) x^{2} x y^{3} $$ Now we apply the product rule to the bases with matching exponents. In this case, it is the terms with the base \(x\). So, using the product rule, \(x^2 \cdot x = x^{2+1} = x^3\). Plug this back into the expression: $$ 32x^3 y^{3} $$

Step 3: State the simplified expression

After applying the product rule of exponents and simplifying the given expression, we get the final simplified expression as: $$ 32x^3 y^{3} $$

Key Concepts

Exponent Simplification

When you're faced with expressions involving exponents, the process to simplify them might seem daunting at first. But fear not! By understanding the product rule of exponents, you can turn a complicated-looking expression into a simpler one with ease.

Let's say we encounter exponents with the same base during multiplication, like in the expression \( a^m \cdot a^n \). The product rule simplifies our task considerably; it tells us to keep the base and add the exponents together, yielding \( a^{m+n} \). Applying this rule to our exercise, we took the base \( x \) with exponents 2 and 1 and added them to get \( x^3 \).

This straightforward rule helps to keep expressions neat and manageable and is a cornerstone in algebra. Remember, the base stays the same, and the exponents do the tango – they simply add up.

Coefficient Multiplication

In algebra, dealing with coefficients is quite similar to performing arithmetic with numbers. When we multiply terms with coefficients, like \(4 \cdot 8 \), we’re essentially doing basic multiplication to combine the numbers. This allows us to condense the expression into a form that reveals only the essential numbers we need.

In the context of our exercise, we multiplied the coefficients \(4\) and \(8\), which gave us \(32\). It's crucial to keep these operations separate from what we do with exponents; coefficients interact with each other through multiplication or division, while exponents follow their own set of rules. Simplifying terms in this way ensures that the coefficients are primed and ready to pair with whatever simplified exponents we've got.

Algebraic Expression Simplification

Simplifying algebraic expressions is akin to cleaning up your room: you organize everything, put like items together, and discard what you don’t need. By using the rules for exponents and coefficients, as we did in our exercise, we transform the jumbled mess of terms into a tidy, simplified expression.

For instance, after applying the rules for coefficient multiplication and exponent simplification, we condensed \(4 x^{2}\) and \(8 x y^{3}\) down to \(32x^3 y^{3}\). We've essentially combined like terms, simplified where possible, and arrived at an expression that is much easier for anyone to understand. This neater form is not only easier on the eyes but also makes it simpler to use in further calculations or when comparing with other expressions.