Problem 61 - Elementary Algebra

Question

Simplify \(\left(x^{4} y^{2} z^{3}\right)^{5}\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(x^{4} y^{2} z^{3}\right)^{5}\). Answer: The simplified expression is \(x^{20} \cdot y^{10} \cdot z^{15}\).

Step 1: Apply the power of a product rule

According to the power of a product rule, we can apply the exponent \((\)5\()\) to each term inside the parentheses: \(\left(x^{4} y^{2} z^{3}\right)^{5} = (x^{4})^{5} \cdot (y^{2})^{5} \cdot (z^{3})^{5}\).

Step 2: Simplify each term

Now, we'll simplify each term, by multiplying the exponents of each variable, according to the rule: \((a^{b})^{c} = a^{bc}\): \((x^{4})^{5} = x^{4 \cdot 5} = x^{20}\), \((y^{2})^{5} = y^{2 \cdot 5} = y^{10}\), and \((z^{3})^{5} = z^{3 \cdot 5} = z^{15}\).

Step 3: Combine the simplified terms

Finally, we'll combine the simplified terms to obtain our final answer: \(x^{20} \cdot y^{10} \cdot z^{15}\).

Key Concepts

Power of a Product Rule

The power of a product rule is a fundamental principle in algebra that comes into play when you're dealing with expressions involving exponents. Simply put, when you have a product of terms raised to an exponent, you can apply that exponent to each term individually. In mathematical terms, this rule states that \( (a \cdot b)^n = a^n \cdot b^n \) for any numbers or variables \( a \) and \( b \) and any positive integer exponent \( n \).

Here's why this works: raising something to an exponent means you're multiplying that something by itself a certain number of times. So, when you have a product—like \( x \cdot y \)—raised to an exponent, you're multiplying that \( x \cdot y \) by itself over and over again. Doing this separately, or 'distributing' the exponent to \( x \) and \( y \) independently, yields the same result. The only catch is that this rule works neatly with multiplication; operations like addition or subtraction require a different approach.

For example, our exercise raised the product \( x^{4} y^{2} z^{3} \) to the fifth power. By applying the power of a product rule, you can simplify this to \( x^{4 \cdot 5} y^{2 \cdot 5} z^{3 \cdot 5} \) without changing the meaning of the original expression.

Exponent Multiplication

When simplifying algebraic expressions, you often encounter exponents that need to be multiplied together. This part of the process is guided by a rule of exponentiation that states when you raise a power to another power, you multiply the exponents. The general form of this exponent multiplication rule is \( (a^b)^c = a^{b \cdot c} \).

So what's happening here? You're taking number \( a \) that has already been multiplied by itself \( b \) times, and then you are multiplying that result by itself \( c \) times. The shortcut is to simply multiply the exponent \( b \) by the new exponent \( c \) right away, which saves time and streamlines the process.

In our sample exercise, we have variables with exponents raised to the fifth power. Using the exponent multiplication rule, \( x^{4} \) raised to the 5th power becomes \( x^{4 \cdot 5} \) or \( x^{20} \) and similarly for the other variables. It's a quick way to calculate the powers without lengthy multiplication.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression into its simplest form, making it easier to understand or further manipulate. The ultimate goal is to make the expression as concise as possible without altering its value. To simplify an algebraic expression, we apply a variety of rules such as combining like terms, factoring, and using the exponent rules we've discussed.

Simplifying an expression often involves recognizing patterns or repeated applications of rules. For instance, when you look at an expression like \( x^{20} \cdot y^{10} \cdot z^{15} \) from the exercise, you know it can't be reduced further because there are no like terms to combine and the exponents are already in their simplest form.

Remember, an algebraic expression is simplified when no more operations can be performed on it. In the context of our previous steps, combining \( x^{20} \) with \( y^{10} \) and \( z^{15} \) using multiplication yields the final simplified product. This doesn't just help with homework - it's a crucial skill for solving complex equations and making informed decisions based on algebraic models.