Problem 40 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(6 x^{4}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression \((6x^4)^2\) is \(36x^8\).

Step 1: Recognizing the problem

Here, the problem presents \((6x^4)^2\). It is noted that the expression inside the brackets is raised to a power of 2. Using the rules of exponents, the power of a product and power of a power rule is applicable here.

Step 2: Applying the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). In this case, \(a\) would be \(6\) and \(b\) would be \(x^4\), while \(n\) would be 2. Hence, the problem becomes \(6^2 * (x^4)^2\).

Step 3: Applying the power of a power rule

The power of a power rule states that \((a^n)^m = a^{nm}\). Here, \(a\) would be \(x\), \(n\) would be 4 and \(m\) would be 2. Hence, the problem again simplifies to \(36 * x^8\).

Step 4: Concluding the problem

Finally, multiplying 36 and \(x^8\) as multiplication is commutative, which means that changing the order of the numbers does not change the product. So, the simplified exponential expression becomes \(36x^8\).

Key Concepts

Power of a Product Rule

When we encounter an expression like \((ab)^n\), where both \(a\) and \(b\) are being raised together to the power of \(n\), we apply the power of a product rule. This rule helps us expand the expression by distributing the exponent over the product inside the parentheses. For example, writing \((6x^4)^2\) under the power of a product implies we need to raise both \(6\) and \(x^4\) to the power of 2 separately.

This means:

• Raise the number \(6\) to the power of 2: \(6^2 = 36\).
• Raise \(x^4\) to the power of 2 using further rules of exponents.

By systematically applying the power of a product, we can simplify complex expressions involving multiple variables or terms into more manageable forms.

Power of a Power Rule

The power of a power rule simplifies expressions where a power is raised to another power, such as \((a^n)^m\). When faced with such a situation, multiply the exponents together to get a single exponent: \(a^{nm}\). This rule drastically simplifies the process of managing expressions with multiple layers of exponents.

Using the example \((x^4)^2\), applying the power of a power rule means we multiply the exponents 4 and 2:

• Calculate \(x^{4 \times 2} = x^8\).

This provides a streamlined way to reduce the computation required with nested exponentials and is particularly useful when dealing with algebraic expressions involving exponents.

Simplifying Expressions

Simplifying expressions in mathematics often involves applying various rules to transform the expression into a more concise form. With exponential expressions, this typically includes using the rules of exponents such as the power of a product and power of a power, as seen in our example.

The original problem \((6x^4)^2\) simplifies to the expression \(36x^8\) by:

• Calculating \(6^2 = 36\) using the power of a product rule for the coefficient.
• Using \((x^4)^2 = x^8\) with the power of a power rule for the variable part.

Finally, by performing these calculations and recombining the results, we achieve the simplified form of the expression, making it easier to work with in further mathematical operations.