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Chapter 2: Problem 81

Compare and contrast the expressions \(3 \cdot 5\) and \(5^{3}\).

Short Answer

Expert verified
15 (\(3 \times 5\)) is much smaller than 125 (\(5^{3}\)). Multiplication and exponentiation are fundamentally different operations, leading to different results.

Step by step solution

01

- Calculate the Value of the First Expression

Evaluate the multiplication expression: \(3 \times 5 = 15\)
02

- Calculate the Value of the Second Expression

Evaluate the power expression: \(5^{3} = 5 \times 5 \times 5 = 125\)
03

- Compare the Results

Compare the values obtained from the previous steps:- The value of \(3 \times 5\) is 15.- The value of \(5^{3}\) is 125.Clearly, 15 is much smaller than 125.
04

- Contrast the Operations

Identify the difference in operations for both expressions:- \(3 \times 5\) involves multiplication of two numbers, resulting in a product.- \(5^{3}\) involves taking the base 5 to the power of 3, resulting in an exponential value.

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Key Concepts

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

Multiplication
Multiplication is one of the four fundamental operations of arithmetic. It represents the process of adding a number to itself a certain number of times. For example, the expression \(3 \times 5\) means you are adding 5 three times, which is 5 + 5 + 5. This results in the product 15.
Understanding multiplication is important because it helps in simplifying complex problems and is used in various aspects of mathematics and everyday life, like calculating areas or understanding repeated addition.
Exponents
Exponents represent repeated multiplication of the same number. The expression \(5^{3}\) means that you multiply 5 by itself three times: \(5 \times 5 \times 5\). This results in 125.
Exponents are a way to express large numbers efficiently. They are crucial in understanding higher-level math topics like algebra, calculus, and even scientific notation. Knowing how to work with and interpret exponents can significantly simplify complex problems.
Comparison of Values
When comparing values derived from different operations like multiplication and exponents, it's essential to look at the results and understand what they represent.
In our example, evaluating \(3 \times 5\) gives 15, while \(5^{3}\) gives 125. Clearly, 125 is much larger than 15. The reason lies in the nature of the operations: multiplication adds a number a specific number of times, while exponents multiply a number by itself a given number of times, growing much faster as the exponent increases.
Comparing values helps in understanding the relative magnitude and the efficiency of different mathematical operations in solving problems.

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Most popular questions from this chapter

Discuss what is wrong (if anything) with each of the following examples: (a) \(5-3(x-4) \stackrel{?}{=} 2(x-4) \stackrel{?}{=} 2 x-8\) (b) \(5-3(x-4) \stackrel{2}{=} 5-3 x-4 \stackrel{2}{=} 1-3 x\) (c) \(5-3(x-4) \geq 5-3 x-12 \frac{2}{2}-7-3 x\) (d) \(5-3(x-4) \geq 5-3 x+12 \leq 17-3 x\)

Evaluate the given expression for \(x=0.24\) and \(y=-0.5 .\) Round off to the nearest thousandth where necessary. $$(x+y)^{2}$$

Examine each of the following expressions as given and determine the number of terms. Also find the coefficient and the literal part of each term. $$4 x^{2}-3 x+2$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$(x+y+z)^{2}$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$\frac{x y z}{z-x}$$
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