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Chapter 7: Problem 33

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

Short Answer

Expert verified
16 r^{12} s^{20}

Step by step solution

01

Understand the Problem

The problem requires simplifying the expression \(\left(2 r^{3} s^{5}\right)^{4}\). This involves raising each component within the parentheses to the power of 4.
02

Apply the Power Rule to Each Term

According to the power rule \((a^m)^n = a^{m\cdot n}\), apply this to each term inside the parentheses: \(2^{4} \, (r^{3})^{4} \, (s^{5})^{4}\).
03

Simplify Each Term

Simplify each term using the power rule: \[ 2^{4} = 16 \ (r^{3})^{4} = r^{12} \ (s^{5})^{4} = s^{20} \].
04

Combine Simplified Terms

Combine the simplified terms into one expression: \(16 r^{12} s^{20}\).

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Key Concepts

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

Power Rule
To simplify expressions where a term is raised to a power, we use the power rule. The power rule states that \((a^m)^n = a^{m \cdot n}\). This means you multiply the exponents.
This rule is very helpful because it lets you simplify expressions quickly.
For example: \((r^3)^4 = r^{3 \cdot 4} = r^{12}\).
Using the power rule, you can simplify expressions like \((a^2 b^3)^3\) quickly to \(a^{2 \cdot 3} b^{3 \cdot 3} = a^6 b^9\).
This method works for all terms, whether they include variables or constants.
Exponents
Exponents are used to show how many times a number, called the base, is multiplied by itself.
For instance, \(r^3\) means \(r \cdot r \cdot r\) (r multiplied by itself 3 times).
Exponents help in writing large numbers in a compact and understandable manner.

Here are some rules for exponents:
  • \textbf{Product rule}: \(a^m \cdot a^n = a^{m+n}\)
  • \textbf{Quotient rule}: \(\frac{a^m}{a^n} = a^{m-n}\)
  • \textbf{Negative exponent rule}: \(a^{-m} = \frac{1}{a^m}\)
When applying these rules, math becomes more manageable. These rules are essential for simplifying algebraic expressions.
Combining Like Terms
Combining like terms is a step in simplifying algebraic expressions. Like terms have the same variables and corresponding exponents, although their coefficients may be different.
For example, in the expression \(3x^2 + 4x^2\), both terms are like terms because they both contain \(x^2\).
To combine these, sum their coefficients: \(3 + 4 = 7\), so \(3x^2 + 4x^2 = 7x^2\).
Combining like terms makes the expression simpler and more readable. It's a fundamental part of algebra that helps in solving equations efficiently and understanding more complex expressions.

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Most popular questions from this chapter

Lamont wants to make two investments: one in the amount of $$ 1400\( at \)9 \%\( interest and a second in the amount of \) 1100 .\( If the annual interest from the two investments is \) 203$, what is the rate of interest for the second investment?

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