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Chapter 7: Problem 34

Simplify each of the following as completely as possible. $$2\left(r^{3} s^{5}\right)^{4}$$

Short Answer

Expert verified
The simplified form is \( 2 r^{12} s^{20} \).

Step by step solution

01

Apply the Power of a Power Rule

First, observe the expression inside the parentheses. According to the power of a power rule \(\big( a^m \big)^n = a^{m \times n}\), raise each component inside the parentheses to the power of 4: \( \big( r^3 s^5 \big)^4 \). Apply the rule: \( r^{3 \times 4} s^{5 \times 4} \).
02

Simplify Exponents

Now, simplify the exponents calculated in Step 1. This gives: \( r^{12} s^{20} \). So, \( \big( r^3 s^5 \big)^4 = r^{12} s^{20} \).
03

Combine with Coefficient

Combine the simplified expression \( r^{12} s^{20} \) with the coefficient 2. Therefore, \( 2 \big( r^3 s^5 \big)^4 = 2 r^{12} s^{20} \).
04

Final Simplified Form

The completely simplified form of the given expression is \( 2 r^{12} s^{20} \).

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Key Concepts

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

Power of a Power Rule
The power of a power rule is a crucial concept in algebra that helps simplify expressions involving exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. The general form of this rule is \(\left(a^m\right)^n = a^{m \times n}\).

For example, in the expression \(\left(r^3 s^5\right)^4\), you apply the power of a power rule to each term inside the parentheses:
  • For \(r^3\), you get \(r^{3 \times 4} = r^{12}\).
  • For \(s^5\), you get \(s^{5 \times 4} = s^{20}\).
So, \(\left(r^3 s^5\right)^4\) simplifies to \(r^{12} s^{20}\). Understanding this rule makes it easier to work with complex expressions involving exponents.
Simplifying Exponents
Simplifying exponents is the next step after applying the power of a power rule. In essence, this means performing the multiplication of the exponents you determined in the previous step.

In our example, after applying the rule, we had \(r^{3 \times 4}\) and \(s^{5 \times 4}\). When we simplify those, we get
  • \(r^{12}\)
  • \(s^{20}\)
So, \(\left(r^3 s^5\right)^4\) translates to \(r^{12} s^{20}\). It's important to be meticulous here because one small mistake in multiplying the exponents can lead to incorrect results.
Coefficients in Algebra
Coefficients are the numerical parts of terms in an algebraic expression. In our example, the number 2 is the coefficient. Coefficients are multiplied by the algebraic terms they accompany.

After simplifying the exponents, the expression was \(r^{12} s^{20}\). Now, we need to combine this result with the initial coefficient, which is 2. This gives us
  • 2 \(r^{12} s^{20}\)
So, applying the coefficient to the product of the powers just means multiplying the coefficient by the simplified algebraic term. The completely simplified final expression becomes \(2 r^{12} s^{20}\). Understanding how to handle coefficients properly is key to solving complex algebraic problems accurately.

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Most popular questions from this chapter

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x+7)(5 x+4)$$

A collection of 40 coins consists of dimes and quarters. If the total value of the coins is \(\$ 6.70,\) how many of each are there?

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a+5)^{2}$$

A car rental agency charges \(\$ 15\) per day plus \(\$ 0.20\) per mile. If a 3 -day rental cost \(\$ 72,\) how many miles were driven?

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you divide 2350 by \(48,600\), the result is closest to (a) 0.05 (b) 0.5 (c) 5 (d) 50 (e) 500
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