Problem 62 - 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. $$ \left(\frac{x^{2} y^{2}}{2 z^{3}}\right)^{5} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(\frac{x^{2} y^{2}}{2 z^{3}}\right)^{5}\). Answer: \(\frac{x^{10} y^{10}}{32 z^{15}}\)

Step 1: Apply exponent to each term inside the parentheses

To simplify the given expression, we will apply the exponent 5 to each term inside the parentheses: \((x^2)^5\), \((y^2)^5\), and \((z^3)^5\). We also need to apply the exponent to the constant 2. $$ \left(\frac{x^{2} y^{2}}{2 z^{3}}\right)^{5} = \frac{(x^{2})^{5} (y^{2})^{5}}{(2)^5 (z^{3})^{5}} $$

Step 2: Use power rules for exponents

Now, we use the power rules for exponents to simplify the expression. When raising a power to another power, we multiply the exponents. Thus, we have: $$ \frac{(x^{2})^{5} (y^{2})^{5}}{(2)^5 (z^{3})^{5}} = \frac{x^{2\times 5} y^{2\times 5}}{2^{5} z^{3\times 5}} $$

Step 3: Simplify exponents

Now, we can simplify the exponents by multiplying them: $$ \frac{x^{2\times 5} y^{2\times 5}}{2^{5} z^{3\times 5}} = \frac{x^{10} y^{10}}{32 z^{15}} $$

Step 4: Write the final simplified expression

After applying the exponent rules, our final simplified expression is: $$ \left(\frac{x^{2} y^{2}}{2 z^{3}}\right)^{5} = \frac{x^{10} y^{10}}{32 z^{15}} $$

Key Concepts

Simplifying Algebraic Expressions

When faced with complex algebraic expressions, the goal is to rewrite them in their simplest form while keeping their values unchanged. Simplifying expressions makes it easier to understand and solve mathematical problems. One common method for simplifying involves the use of exponent rules, such as the power rule, which plays a crucial role in reducing expressions that contain powers or exponents.

An expression like \(\left(\frac{x^{2} y^{2}}{2 z^{3}}\right)^{5}\) may seem daunting at first glance, but by breaking it down and applying the power rule, we can systematically reduce it to a simpler form. Remember, in simplifying, we are not changing the value, just the appearance, making it more comprehensible and manageable for further operations or evaluations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. It's a shorthand for repeated multiplication. For example, \(x^{2}\) means \(x \times x\). The rules of exponents are shortcuts that help us manipulate expressions with exponents efficiently.

One of these key principles is the power rule, which states that when taking a power of a power, you multiply the exponents. For instance, \( (x^a)^b = x^{a \times b} \). This concept is integral in simplifying the given textbook exercise, turning a seemingly intricate expression into a more straightforward one by multiplying the exponents when the power outside the parentheses is applied.

Algebraic Manipulation

Algebraic manipulation refers to the various techniques used to rewrite expressions and equations in different forms. In this context, manipulation may involve applying exponent rules, distributing factors, combining like terms, and using properties of operations to reshape algebraic expressions into forms that are easier to work with or solve.

In our example, manipulation begins with applying the external exponent to every component within the parentheses, including both the variables and the constant. This type of manipulation is the cornerstone of algebra, and practicing these methods builds a solid foundation that is applicable to solving more advanced mathematical problems.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and increasing by 1 each time (1, 2, 3, ...). They represent countable amounts and are fundamental in many areas of mathematics. In an algebraic context, exponents are often assumed to be natural numbers unless stated otherwise. This assumption keeps the operations and results well-defined and avoids the complications that can arise with zero or negative exponents or fractional exponents.

The exercise specifies that all variable exponents are natural numbers, which simplifies our steps and the application of the power rules. Understanding these foundational number sets is crucial as they lay the groundwork for grasping more abstract mathematical concepts and for ensuring the proper application of rules and operations pertinent to the study of algebra.