Problem 66 - Brief 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 Power Rule for Exponents

First, simplify the expression inside the parentheses by using the power rule for exponents, which states that when multiplying like bases, you add the exponents. In this case, the like bases are both 'x'. So add the exponents 4 and 3: \(x^4 \cdot x^3 = x^{4+3} = x^7\).

Step 2: Apply the Power of a Power Rule

Now apply the power of a power rule to the entire expression \((x^7)^2\),which means you multiply the exponents: \((x^7)^2 = x^{7 \cdot 2} = x^{14}\).

Step 3: Write the Final Simplified Expression

After applying the rules, you arrive at the final simplified expression. Therefore, \((x^4 \cdot x^3)^2\) simplifies to \(x^{14}\).

Key Concepts

Power Rule for Exponents

The power rule for exponents is an essential concept when dealing with expressions that have like bases. It states that when you multiply two algebraic terms with the same base, you simply add their exponents. This is symbolically represented as:

• \(a^m\cdot a^n = a^{m+n}\)

In our example, we have \(x^4\) and \(x^3\) being multiplied inside the parentheses. Both these terms share the same base 'x'. So, according to the power rule for exponents, we add the exponents 4 and 3. This simplifies to:

• \(x^4 \cdot x^3 = x^{4+3} = x^7\)

Using this method, we reduce the complexity by focusing solely on the exponents, making it easier to handle larger terms. Remember, this rule only applies when the bases are identical.

Power of a Power Rule

The power of a power rule comes into play when you take an exponential expression and raise it to another power. This rule dictates that you multiply the exponents. The general form is:

• \((a^m)^n = a^{m\cdot n}\)

In our exercise, after simplifying the inner pair \((x^4 \cdot x^3)\) to \(x^7\), we then raise this term to the power of 2. Applying the power of a power rule means multiplying the exponent of the expression by the external power:

• \((x^7)^2 = x^{7 \cdot 2} = x^{14}\)

This step is crucial for condensing expressions further and helps maintain order when faced with nested exponentials.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to read and work with. The process requires understanding and applying the rules of exponents correctly to consolidate terms. In our scenario, we started with a more complex expression:

• \((x^4 \cdot x^3)^2\)

By using the rules for exponents, we transformed this expression into a much simpler form, \(x^{14}\). Simplifying like this:

• Combines like terms, reducing clutter.
• Makes subsequent mathematical operations more manageable.
• Helps to clearly identify core terms and coefficients.

A simplified expression not only looks cleaner but also allows for quicker problem solving in various contexts of mathematics.