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Chapter 5: Problem 25

Use properties of exponents to simplify each expression. First express the answer in exponential form. Then evaluate the expression. \(3^{4} \cdot 3^{-2}\)

Short Answer

Expert verified
The simplified version of \(3^{4} \cdot 3^{-2}\) is \(3^{2}\) which evaluates to 9.

Step by step solution

01

Understand the Problem

The given expression is \(3^{4} \cdot 3^{-2}\). We are supposed to simplify this expression. Then, after getting the simplification in exponential form, we must also evaluate it.
02

Apply Exponent Rules

Applying the rule of exponents, which states that when you are multiplying expressions with the same base, you add the exponents, get: \(3^{4-2}\).
03

Simplify the Expression

After subtracting the exponents, we get \(3^{2}\). This is the simplification in exponential form.
04

Evaluate the Simplified Expression

Now we evaluate \(3^{2}\). The power (or exponent) of '2' means that the base (3) is used as a multiplier twice. So, \(3^{2}\) is equal to \(3 \times 3 = 9\).

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Key Concepts

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

Exponential Form
Exponential form is a way of expressing numbers using a base and an exponent. This is particularly useful when dealing with very large or very small numbers as it simplifies the expression. For example, in the expression \(3^{4}\), 3 is the base and 4 is the exponent. It tells us that 3 should be multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3\). Converting a given problem into exponential form is often the first step, as it helps in understanding the structure of the expression. In our original exercise, the expression is already in exponential form, \(3^{4} \cdot 3^{-2}\), indicating we have a consistent base of 3 that simplifies our calculations.
Exponent Rules
Understanding exponent rules is essential in simplifying expressions like the one we have. One fundamental rule states that when multiplying two exponential expressions with the same base, you can add the exponents. This rule is expressed as: \[a^{m} \cdot a^{n} = a^{m+n}\]In the given problem, \(3^{4} \cdot 3^{-2}\), both terms have the base of 3. By applying the exponent rule, we combine the exponents by adding them: \[4 + (-2) = 2\]This simplifies the expression to \(3^{2}\). Mastering these exponent rules allows for straightforward simplification and manipulation of terms with exponents.
Simplifying Expressions
Simplifying expressions involves reducing them to a more compact or easy-to-use form while maintaining equivalence. Once you apply the rules of exponents, as seen in the previous section, you can achieve a simpler expression. In our exercise, after applying the exponent rule, the original expression \(3^{4} \cdot 3^{-2}\) simplifies to \(3^{2}\). This reduced form makes it much easier to evaluate the expression or apply further operations if necessary.Whenever simplifying, always ensure that every step adheres to mathematic laws to maintain the correctness and integrity of the solution.
Evaluating Expressions
Once we have simplified an expression, evaluating it becomes the next step. Evaluating means calculating the numeric value of the expression. For \(3^{2}\), it means determining what number \(3 \cdot 3\) yields.Here's how you evaluate \(3^{2}\):
  • The base is 3, and the power of 2 means we multiply 3 by itself.
  • So, \(3 \cdot 3 = 9\).
Thus, \(3^{2}\) evaluates to 9. Simplification followed by evaluation provides both a concise expression and a numeric solution, making complex expressions manageable and easier to understand.

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Most popular questions from this chapter

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(3,15,75,375, \ldots\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,-6,-18,-30, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=18, r=-\frac{1}{3}\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.
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