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Chapter 7: Problem 10

Simplify each of the following as completely as possible. $$3^{2} 5^{3}$$

Short Answer

Expert verified
The answer is 1125.

Step by step solution

01

- Evaluate each exponent separately

Start by finding the value of each base raised to its exponent. For the term \(3^{2}\), calculate \(3 \times 3\). For the term \(5^{3}\), calculate \(5 \times 5 \times 5\).
02

- Compute \(3^{2}\)

\(3^{2} = 3 \times 3 = 9\)
03

- Compute \(5^{3}\)

\(5^{3} = 5 \times 5 \times 5 = 125\)
04

- Multiply the results

Now multiply the results from Step 2 and Step 3: \(9 \times 125\).
05

- Calculate the final result

\(9 \times 125 = 1125\)

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is crucial in mathematics. It involves reducing complex expressions into simpler forms, making it easier to understand and solve. In this exercise, you are asked to simplify the expression \(3^{2} 5^{3}\). The goal is to break this expression down into simpler parts and then combine them to find the final result.

We start by examining the individual components of the expression. Each part needs evaluation separately before combining the results. Knowing how to simplify expressions makes solving problems quicker and more accurate, aiding in achieving better results in math.
Multiplication
Multiplication is one of the basic arithmetic operations, where you combine two or more numbers to find a total. In this exercise, multiplication occurs at two stages:
  • When raising each base to its exponent
  • When combining the results of those exponents
First, for \(3^{2}\), multiplication is repeated two times: 3 multiplied by 3. Similarly, for \(5^{3}\), multiplication is repeated three times: 5 multiplied three times. After these calculations, the results (9 and 125) are multiplied together.

Understanding multiplication and its role in exponentiation helps in breaking down complex problems and avoids errors.
Base and Exponent
A base and an exponent are the two components of an exponential expression. The base tells you what number is being multiplied, and the exponent tells you how many times the base is used as a factor. In \(3^{2}\), 3 is the base, and 2 is the exponent. This means you multiply 3 by itself two times. Similarly, in \(5^{3}\), 5 is the base, and 3 is the exponent, indicating that 5 is multiplied by itself three times.
  • Base: The number being multiplied
  • Exponent: The number of times you multiply the base by itself
Knowing this relationship simplifies the calculation process, enabling you to tackle exponentiation with clarity and ease.

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