Problem 21 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(5 p^{10}\right)^{3}$$

Step-by-Step Solution

Answer

\(125p^{30}\)

Step 1: Apply the power of a product rule

To simplify \((5p^{10})^3\), we will first apply the power of a product rule. The power of a product rule states that the power raised to a product can be written as the product of the individual powers. In this case, we have: \((5p^{10})^3 = 5^3 (p^{10})^3\)

Step 2: Apply the power of a power rule

Now we can apply the power of a power rule, which states that a power raised to another power equals the product of the powers. Applying this rule to the exponent of the variable p in our expression: \(5^3 (p^{10})^3 = 5^3 p^{(10)(3)}\)

Step 3: Solve the exponents

With the power of a power rule applied, we can now simplify the exponents. First, multiply the exponents of the variable p: \(5^3 p^{(10)(3)} = 5^3 p^{30}\) Then, solve the exponent for the constant 5: \(5^3 p^{30} = 125 p^{30}\)

Step 4: Write the final simplified expression

Our expression has been simplified, and there are no negative exponents. The final simplified expression is: \(125p^{30}\)

Key Concepts

Power of a Product Rule

When simplifying expressions like \((5p^{10})^3\), the power of a product rule is a crucial tool. This rule allows us to distribute the exponent across each factor inside the parentheses. In practical terms:

• The expression \((ab)^n\) becomes \(a^n b^n\).

For example, \((5p^{10})^3\) is simplified to \(5^3 (p^{10})^3\). This rule ensures that each factor, both numerical and variable, is raised to the power separately, making the simplification process more straightforward. By doing so, we maintain the same mathematical value while simplifying the structure.

Power of a Power Rule

The power of a power rule is applied when you have an exponent that is raised to another exponent. This can be written generally as:

• \((a^m)^n = a^{m \cdot n}\)

In our exercise, we see this rule in action with \((p^{10})^3\). Applying the rule, we multiply the exponents: \(10 \cdot 3\). This transforms it into \(p^{30}\). It's a straightforward concept, yet incredibly useful when handling complex expressions where variables are raised to powers, which themselves are also raised to higher powers.

Simplifying Expressions

After applying the rules, the final step involves computation and combination of like terms. In our example, we end up with \(5^3 p^{30}\). We calculate \(5^3\), which is equivalent to 125, giving us the expression:

• \(125p^{30}\)

This represents the fully simplified version of the original expression \((5p^{10})^3\). Simplifying expressions often involves identifying patterns and applying exponent rules correctly. The process requires attention to detail to ensure all calculations are accurate and the final result does not contain negative exponents.