Problem 59 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(x^{2} x^{3}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{25} \).

Step 1: Apply the Power Rule of Exponents

First, recall that the power rule of exponents states that \( (a^m)^n = a^{m \cdot n} \). We need to apply this rule to the entire expression. Currently, the expression is inside parentheses and has an outer exponent of 5: \( \left(x^2 x^3\right)^5 \). We will address the inside of the parentheses later. For now, apply the rule as follows: \[ \left(x^2 x^3\right)^5 = (x^2)^5 (x^3)^5 \] Simplifying this gives us \[ x^{10} x^{15} \]

Step 2: Apply the Product Rule of Exponents

Next, recall that the product rule of exponents states \( a^m \cdot a^n = a^{m + n} \). We will apply the product rule to the terms we obtained in the previous step: \((x^{10})(x^{15})\).Using the product rule, add the exponents:\[ x^{10 + 15} = x^{25} \] This simplifies the expression to one single term.

Key Concepts

Product Rule of Exponents

The Product Rule of Exponents is incredibly handy when multiplying terms that have the same base. It tells us that when we multiply two exponents with the same base, we can simply add their exponents together. This rule can be expressed as \( a^m \cdot a^n = a^{m + n} \).
In our original exercise, once the expression inside the parentheses is simplified, you apply the product rule to combine terms like \(x^{10}\) and \(x^{15}\).
By adding the exponents, we get \(x^{25}\).

• Identify the base that is common in the terms you are multiplying.
• Add up the exponents for those terms.
• The result is a single term where the base is raised to the power of the sum of the exponents, simplifying your expression.

Power Rule of Exponents

The Power Rule of Exponents helps us when a power is raised to another power. In these situations, you multiply the exponents. This is written as \((a^m)^n = a^{m \cdot n}\).
In our exercise, we needed to initially focus on the expression inside the parentheses \((x^2 x^3)^5\).
Applying the Power Rule is like distributing the outer exponent to each term inside the parentheses. So, each term gets raised to the power.
The steps were:

• Apply the power to each term: \((x^2)^5 \times (x^3)^5\).
• Multiply the exponents: \(x^{2\cdot 5}\) becomes \(x^{10}\), and \(x^{3\cdot 5}\) becomes \(x^{15}\).
• This simplifies the expression to \(x^{10} \times x^{15}\).

Simplifying Expressions

Simplifying expressions is all about reducing them to the most straightforward form. By applying the rules of exponents, you can transform complex expressions into much simpler ones.
With problems involving exponents, approach simplifying in these steps:

• Apply the Power Rule first to handle any power-over-parentheses situations.
• Use the Product Rule to combine any terms with the same base.
• Combine like terms and make sure everything's consolidated into the simplest form.

The goal is to rewrite the expression so that it is easier to understand and work with. It can involve a combination of several operations, like adding exponents through the product rule or multiplying them as per the power rule. Seeing how these operations transform the expression helps in understanding the structure and behavior of algebraic expressions.