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Chapter 5: Problem 150

Simplify each expression. $$ \left(10 p^{4}\right)^{3}\left(5 p^{6}\right)^{2} $$

Short Answer

Expert verified
25000 \cdot p^{24}

Step by step solution

01

- Apply the Power Rule to Each Term

First, apply the power rule \( a^m \cdot n = a^{mn}\) to each term separately. \((10 p^4)^3 \) becomes \(10^3 \cdot (p^4)^3 = 1000 \cdot p^{12}\). \((5 p^6)^2 \) becomes \(5^2 \cdot (p^6)^2 = 25 \cdot p^{12}\)
02

- Multiply the Numerical Coefficients

Multiply the numerical coefficients from \( (1000 \cdot p^{12}) \) and \( (25 \cdot p^{12}) \). This results in \(1000 \cdot 25 = 25000\)
03

- Add the Exponents of Like Bases

Add the exponents of the like bases \(p^{12} \) and \(p^{12} \). Using the rule \(a^m \cdot a^n = a^{m+n} \), this results in \(p^{12+12} = p^{24}\)
04

- Combine the Results

Combine the numerical coefficient and the simplified base with its exponent to get the final answer: \( 25000 \cdot p^{24}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
When simplifying algebraic expressions, the Power Rule is very useful. This rule states that when you raise a power to another power, you multiply the exponents. The formula is \((a^m)^n = a^{mn}\). Let's look at an example from the exercise: \( (10 p^4)^3 \). Using the Power Rule, you first raise the numerical coefficient and each individual variable inside the parenthesis to the third power. This results in: \[ (10)^3 \times (p^4)^3 = 1000 \times p^{12} \] Likewise, for the second part \( (5 p^6)^2 \), applying the Power Rule gives: \[ (5)^2 \times (p^6)^2 = 25 \times p^{12} \] By mastering the Power Rule, you simplify complex expressions much more efficiently.
Exponents Addition
The rule for adding exponents is crucial when multiplying terms with the same base. The rule is \( a^m \times a^n = a^{m+n} \). After applying the Power Rule, we end up with two terms: \[ 1000 p^{12} \] and \[ 25 p^{12} \]. Notice that both terms have the same base, \( p \). To simplify, we add up the exponents: \[ p^{12} \times p^{12} = p^{12+12} = p^{24} \] You do not add the coefficients, only the exponents of like bases. This step is key to simplification.
Multiplication of Coefficients
The next important step is multiplying the numerical parts of the terms. After using the Power Rule, we have coefficients 1000 and 25. Multiplying these gives: \[ 1000 \times 25 = 25000 \] This simplifies the original complex expression by combining all numerical multipliers together. Finally, combine this with the simplified base with its exponent: \[ 25000 \times p^{24} \] So, the final expression becomes: \[ 25000 p^{24} \] Understanding how to correctly multiply coefficients ensures that you don't miss out critical parts in simplifying algebraic expressions.

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