Problem 26 - 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. $$ (2 x y)^{5} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2xy)^5\). Answer: The simplified expression is \(32x^5y^5\).

Step 1: Apply the power of a product rule

To apply the power of a product rule, remember that \((ab)^n = a^n b^n\). Therefore, we can rewrite the expression as follows: $$ (2xy)^5 = 2^5 \cdot x^5 \cdot y^5 $$

Step 2: Calculate the numerical exponent

Simplify the numerical exponent for 2 which is raised to the power of 5: $$ 2^5 = 32 $$

Step 3: Combine the simplified terms

Now that we have the simplified numerical exponent, we can replace \(2^5\) with \(32\): $$ 32 \cdot x^5 \cdot y^5 $$

Step 4: Write the final simplified expression

The final simplified expression based on power rules for exponents is: $$ 32x^5y^5 $$

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in algebra which makes equations and functions easier to work with. Simplification can include a variety of methods such as combining like terms, factoring, expanding expressions, and utilizing exponent rules.

In the case of expressions involving exponents, simplification often involves applying specific rules that define how to handle powers of products or quotients. For example, when an entire product is raised to an exponent, the power rule states that each base in the product is raised to the exponent separately. This is what we used in our exercise to simplify \( (2 x y)^{5} \). The process involves breaking down the expression into smaller, more manageable parts, which are then individually raised to the given power and subsequently recombined to give a simplified form.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied by itself, and the exponent tells us how many times the base is used in the multiplication. So, if we have \(2^5\), this means 2 is multiplied by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

The power rules for exponents are shortcuts that help you simplify expressions without having to multiply the base by itself over and over again. One such rule is the power of a product rule, highlighted in the exercise, which allows us to take a product raised to an exponent and apply the exponent to each factor individually, greatly simplifying the process of calculating large powers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific mathematical quantity. Unlike equations, they don't have an equals sign; instead, they are phrases that can be simplified or manipulated according to algebraic rules.

Working with algebraic expressions requires an understanding of variables, which are symbols used to represent unknown values. The expression \(32x^5y^5\), for instance, shows a relationship between the variables x and y, each raised to the fifth power, and multiplied by the constant 32. When you encounter mixed terms with coefficients, variables, and exponents, the rules for exponentiation come into play to simplify these expressions efficiently, as demonstrated in the provided exercise.