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Chapter 5: Problem 120

Explain the power rule for exponents. Use \(\left(3^{2}\right)^{4}\) in your explanation.

Short Answer

Expert verified
Applying the power rule to our example \((3^{2})^{4}\) gives us \(3^{(2*4)} = 3^{8} = 6561\).

Step by step solution

01

Identify the Base and Exponents

First, the given expression is \((3^{2})^{4}\). In this expression, 3 is the base, and 2 and 4 are the exponents.
02

Apply the Power Rule for Exponents

According to the power rule for exponents, when a power is raised to another power, you multiply the exponents together. Applying this rule to our expression would result in a new expression: \(3^{(2 * 4)}\).
03

Simplify the Exponents

Multiply the exponents together. This results in \(3^{8}\), which is equivalent to the original expression \((3^{2})^{4}\).
04

Calculate the Final Answer

Calculate \(3^{8}\), the result is 6561.

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. They consist of two parts: the base and the exponent. When you see something like \(3^2\), the base is 3, and the exponent is 2. This means that you multiply 3 by itself, which gives you \(3 \times 3 = 9\).

Exponents are helpful in simplifying expressions, making calculations with large numbers easier to manage. They also help in comparing the growth of numbers in different mathematical contexts. Moreover, they are widely used in various fields such as science and engineering.
Base and Exponents Relationship
The base in an expression with exponents is the number being multiplied, while the exponent tells us how many times the base is used as a factor. For example, in the expression \((3^{2})^{4}\), the base is 3. However, in this case, 3 is first raised to the power of 2, which in turn becomes the base for the next exponent, 4.

When we apply the power rule, the exponents themselves are manipulated: they are multiplied with each other. Therefore, in \((3^2)^4\), both exponents 2 and 4 are multiplied, transforming the expression into \(3^{8}\). This approach is not just convenient but also essential in simplifying more complex expressions involving exponents.
Simplifying Expressions Using the Power Rule
Simplifying expressions with exponents becomes straightforward when you understand the power rule. The power rule for exponents states that when you raise an exponential expression to another power, you simply multiply the exponents. This means \((a^m)^n = a^{m \times n}\).

Applying this to the expression \((3^2)^4\):
  • First, identify the exponents: here they are 2 and 4.
  • Next, multiply these exponents together, resulting in \(2 \times 4 = 8\).
  • Then, rewrite the expression as \(3^8\).
This new expression is simpler to calculate when evaluating or embedding in larger equations. Ultimately, performing these simplifications can greatly aid in both mental calculations and more advanced mathematical operations.

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Most popular questions from this chapter

What is an arithmetic sequence? Give an example with your description.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-1000, r=0.1\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(5,15,45,135, \ldots\)

Use the appropriate formula shown above to find \(2+4+6+8+\cdots+200\), the sum of the first 100 positive even integers.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=3\)
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