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Chapter 0: Problem 1

In Exercises \(1-28\), compute the numbers. $$ 3^{3} $$

Short Answer

Expert verified
27

Step by step solution

01

Identify the Base and Exponent

The given expression is \(3^3\). Here, the base is 3 and the exponent is 3. The exponent tells us how many times to multiply the base by itself.
02

Multiply the Base by Itself

Multiply the base, 3 by itself, three times: \(3 \times 3 \times 3\).
03

Perform the Multiplication

First, calculate \(3 \times 3\), which equals 9. Then, multiply 9 by 3 to get the final result: \(9 \times 3 = 27\).
04

State the Final Answer

The value of \(3^3\) is 27.

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Key Concepts

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

mathematical operations
Mathematical operations are the basic building blocks of arithmetic. They allow us to manipulate numbers in various ways to achieve a desired outcome. The primary ones include addition, subtraction, multiplication, and division. In this exercise, we're focusing on exponentiation, which is an advanced form of multiplication. It's crucial to understand each operation to tackle more complex mathematical problems later. Breaking down these operations step-by-step helps build a strong foundation in arithmetic. Let鈥檚 explore the specifics of this operation and how it relates to the given example involving exponentiation.
base and exponent
When dealing with exponentiation, we encounter two important parts: the base and the exponent. In the expression \(3^3\), 3 is the base and 3 is the exponent. The base is the number being multiplied, while the exponent indicates how many times the base is multiplied by itself. In simpler terms, \3^3\ means multiplying 3 by itself three times. This operation can be written as \(3 \times 3 \times 3\). Understanding this relationship is fundamental to grasping how exponentiation works. It simplifies large multiplicative operations into a more manageable form.
multiplication
Multiplication helps in combining groups of equal sizes. When working with exponentiation, think of multiplication as repeated addition. In our exercise, performing \(3^3 \) means we need to multiply 3 by itself repeatedly. First, we calculate \3 \times 3\, which equals 9. Then, we take the result, 9, and multiply it by 3 again: \(9 \times 3 = 27\). This orderly process ensures accuracy and helps in understanding how large numbers are constructed from smaller ones. Mastering basic multiplication is essential for more complex arithmetic operations and for effective problem solving.

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Most popular questions from this chapter

During the first \(\frac{1}{2}\) hour, the employees of a machine shop prepare the work area for the day's work. After that, they turn out 10 precision machine parts per hour, so the output after \(t\) hours is \(f(t)\) machine parts, where \(f(t)=10\left(t-\frac{1}{2}\right)=10 t-5, \frac{1}{2} \leq t \leq 8\). The total cost of producing \(x\) machine parts is \(C(x)\) dollars, where \(C(x)=.1 x^{2}+25 x+200\). (a) Express the total cost as a (composite) function of \(t\). (b) What is the cost of the first 4 hours of operation?

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}}\). Calculate the following functions. Take \(x>0\). \(g(f(x))\)

A cellular telephone company estimates that, if it has \(x\) thousand subscribers, its monthly profit is \(P(x)\) thousand dollars, where \(P(x)=12 x-200\). (a) How many subscribers are needed for a monthly profit of 160 thousand dollars? (b) How many new subscribers would be needed to raise the monthly profit from 160 to 166 thousand dollars?

Annual Compound with Deposits Assume that a couple invests \(\$ 4000\) each year for 4 years in an investment that earns \(8 \%\) compounded annually. What will the value of the investment be 8 years after the first amount is invested?

Annual Compound Assume that a couple invests \(\$ 1000\) upon the birth of their daughter. Assume that the investment earns \(6.8 \%\) compounded annually. What will the investment be worth on the daughter's 18 th birthday?
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