Problem 72 - Applied Calculus

Question

Simplify. $$ \left(x^{4} \cdot x^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{14} \).

Step 1: Apply the Product of Powers Rule

The expression inside the parentheses is \( x^4 \cdot x^3 \). According to the Product of Powers Rule, when you multiply two expressions with the same base, you add the exponents: \( x^a \cdot x^b = x^{a+b} \). So, \( x^4 \cdot x^3 = x^{4+3} = x^7 \).

Step 2: Apply the Power of a Power Rule

Now consider \( (x^7)^2 \). To simplify an expression raised to a power from another power, use the Power of a Power Rule \( (x^m)^n = x^{m\cdot n} \). Thus, \( (x^7)^2 = x^{7\cdot2} = x^{14} \).

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental concept in exponentiation. This rule helps in simplifying expressions where two expressions with the same base are multiplied. When you see something like \( x^a \cdot x^b \), the Product of Powers Rule tells us to add the exponents together. It simplifies to \( x^{a+b} \).

Consider the expression \( x^4 \cdot x^3 \). Both parts have the base \( x \), and thus, we can apply the rule. Just add the exponents: \( 4 + 3 \). This gives us \( x^7 \).

The beauty of this rule is that it streamlines complex expressions into more manageable forms. It saves time and reduces the potential for mistakes in larger calculations.

Power of a Power Rule

Another crucial exponentiation rule is the Power of a Power Rule. This rule is used when you have an expression raised to another power. In mathematical terms, if \( x^m \) is raised to another exponent \( n \), it will simplify according to the formula \( (x^m)^n = x^{m \cdot n} \).

Let's observe the expression \( (x^7)^2 \). The base, \( x^7 \), and the exponent, 2, interact by multiplying the exponents: \( 7 \cdot 2 \). This results in \( x^{14} \).

Using the Power of a Power Rule makes it easy to handle expressions involving multiple layers of exponents. This rule emphasizes the multiplicative nature of exponentiation, offering a simple route to break down complex problems.

Simplifying Expressions

Simplifying expressions is a fundamental skill in working with algebraic operations. It involves reducing expressions to their simplest form to make calculations more straightforward. When applying exponentiation rules, simplification becomes even more intuitive.

In this exercise, we dealt with the expression \( \left(x^{4} \cdot x^{3}\right)^{2} \). To simplify:

• First, use the Product of Powers Rule: \( x^4 \cdot x^3 = x^7 \).
• Next, apply the Power of a Power Rule on \( (x^7)^2 \) to get \( x^{14} \).

Each step distills the expression down to its most basic terms, helping avoid errors in larger calculations. By mastering these simplification techniques, you can tackle more complicated math problems with confidence.