Problem 35 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(x^{2} y^{3} z^{5}\right)^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((x^{2} y^{3} z^{5})^{4}\) using the power rules for exponents. Answer: \(x^8 y^{12} z^{20}\)

Step 1: Rewrite the expression using power rules

We will rewrite the expression using exponentiation rule: \((a^m)^n = a^{mn}\). $$ \left(x^{2} y^{3} z^{5}\right)^{4} $$

Step 2: Apply the power rules to each variable exponent

Apply the exponentiation rule to each variable inside the parentheses: $$ (x^{2\cdot 4}) (y^{3\cdot 4}) (z^{5\cdot 4}) $$

Step 3: Simplify the exponents

Now, we will simplify the exponents $$ (x^{8}) (y^{12}) (z^{20}) $$

Step 4: Combine the variables

After simplifying the exponents, we can write the expression by combining the variables: $$ x^8 y^{12} z^{20} $$

Key Concepts

Simplifying Exponents

Exponents express repeated multiplication and can often seem daunting at first glance. However, simplifying exponents is a matter of understanding and applying a set of rules.

Take the expression \( (x^{2} y^{3} z^{5})^{4} \). Simplifying this expression means applying the power rule of exponents, which states that \( (a^m)^n = a^{mn} \). This rule empowers us to deal with each variable and its exponent separately, maintaining the base and multiplying the exponents.

To visually simplify the expression, we can treat each base with its outside exponent individually: \(x^{2}\) becomes \(x^{2*4}\), resulting in \(x^{8}\). Similarly, \(y^{3}\) becomes \(y^{3*4}\), which simplifies to \(y^{12}\), and \(z^{5}\) yield \(z^{5*4}\), simplifying down to \(z^{20}\).

Breaking down the process in this step-by-step manner turns a complex expression into a more manageable form. Always ensure that the base is non-zero and the exponents are natural numbers to apply this rule effectively.

Exponentiation Rule

Understanding the exponentiation rule is key to working with powers in algebra. It's essentially a shortcut for simplifying expressions with multiple layers of exponents. This rule tells us what to do when we have a power of a power, that is, when an exponent is raised to another exponent.

In the context of our example \( (x^{2} y^{3} z^{5})^{4} \), the exponentiation rule is what allows us to distribute the outer exponent of 4 to each of the inner exponents. Mathematically, this rule is expressed as \( (a^m)^n = a^{mn} \).

Take the base \( x \), raised to the power of 2, and then raised to the power of 4: according to the exponentiation rule, we multiply the exponents 2 and 4 to get the new exponent, which is 8. No matter how large the numbers or complicated the expression, this rule remains consistent and applicable. It’s a fundamental concept that makes dealing with exponential expressions significantly simpler.

Algebraic Expressions

Algebraic expressions are the bread and butter of algebra, combining numbers, variables, and arithmetic operations to represent relationships and solve problems.

In an algebraic expression like \( x^8 y^{12} z^{20} \), we are looking at a simplified version of a previously more complex expression involving exponents. Through the application of exponent rules, we simplify the expression to make it easier to understand and use in further calculations.

Understanding how to manipulate these expressions is fundamental to algebra. It involves knowing how to add, subtract, multiply, divide, and simplify using the rules of exponents. Algebraic expressions can model real-world problems and are essential in finding solutions to such problems. Their power lies in their ability to represent a wide range of situations in a precise, concise manner, from the growth of investments over time to the calculation of distances using the laws of physics.