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Chapter 10: Problem 46

Use a calculator to find each product. Round to the nearest tenth. $$3.14(2.4)^{2}$$

Short Answer

Expert verified
The product is approximately 18.1.

Step by step solution

01

Calculate the Exponent

First, we need to square the number in the parentheses: 2.4. Calculate: \(2.4^2 = 2.4 \times 2.4 = 5.76\).
02

Multiply with the Coefficient

Now, multiply the result from Step 1 by the coefficient 3.14.Calculate: \(3.14 \times 5.76 = 18.0864\).
03

Round the Product

Round the product from Step 2 to the nearest tenth. The number is 18.0864; observe the digit in the hundredths place (8). Since it is equal to or greater than 5, round up the tenths place by 1. Thus, 18.0864 rounds to 18.1.

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Key Concepts

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

Rounding Numbers
Rounding numbers is a mathematical process that simplifies a number to make it easier to work with. The critical part is deciding which digit to round to. In this instance, we round to the nearest tenth.
Take the number you wish to round. Identify the tenths place and the digit immediately next to it, which is in the hundredths place.
  • If the digit in the hundredths place is 5 or higher, increase the tenths place digit by 1.
  • If the hundredths digit is less than 5, keep the tenths digit unchanged.
For example, in 18.0864, the tenths place is the zero just after the decimal, and the hundredths place is 8. Since 8 is greater than 5, we round the tenths place up creating 18.1. Rounding ensures numbers are easier to handle while maintaining accuracy for practical purposes.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It represents repeated multiplication of the base. Here, we have the expression \(2.4^2\), meaning 2.4 is multiplied by itself.
To calculate \(2.4^2\):
  • Multiply: 2.4 \(\times\) 2.4. This basic multiplication step involves arranging the figures and calculating as you would in standard multiplication.
  • Result: The answer from this multiplication gives 5.76, indicating how many times 2.4 multiplies by itself.
Exponentiation is key in various fields like physics, statistics, and finance, as it simplifies handling larger computations efficiently.
Multiplication
Multiplication is a basic arithmetic operation essential for solving various real-world problems. It involves finding the product of two or more numbers, often called the factors.
In problem-solving, when you encounter multiplication:
  • Identify the numbers or variables to be multiplied, called multiplicands.
  • Perform the multiplication calculation accurately for precise outputs.
In our example, you first square 2.4, getting 5.76.
Next step is multiplying 3.14 with 5.76. The process involves calculating \(3.14 \times 5.76 = 18.0864\).
Using a calculator helps, especially with bigger numbers or decimal multiplication, reducing errors and speeding up processes. Multiplication as a math operation is not only foundational but also broad-reaching, necessary in day-to-day activities as well as in advanced mathematics.

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