Question

Write each of the following numbers as "ordinary" decimal numbers. a. \(6.244 \times 10^{3}\) b. \(9.117 \times 10^{-2}\) c. \(8.299 \times 10^{1}\) d. \(1.771 \times 10^{-4}\) e. \(5.451 \times 10^{2}\) f. \(2.934 \times 10^{-5}\)

Short answer

a. 6244 b. 0.09117 c. 82.99 d. 0.0001771 e. 545.1 f. 0.00002934

Step-by-step solution

a. 6.244 × 10^3

To convert this number, we will multiply 6.244 by 10 raised to the power of 3. That is \(6.244 \times 10^3 = 6.244 \times 1000\), which results in the decimal number 6244.

b. 9.117 × 10^(-2)

To convert this number, we will multiply 9.117 by 10 raised to the power of -2. That is \(9.117 \times 10^{-2} = 9.117 \times 0.01\), which results in the decimal number 0.09117.

c. 8.299 × 10^1

To convert this number, we will multiply 8.299 by 10 raised to the power of 1. That is \(8.299 \times 10^1 = 8.299 \times 10\), which results in the decimal number 82.99.

d. 1.771 × 10^(-4)

To convert this number, we will multiply 1.771 by 10 raised to the power of -4. That is \(1.771 \times 10^{-4} = 1.771 \times 0.0001\), which results in the decimal number 0.0001771.

e. 5.451 × 10^2

To convert this number, we will multiply 5.451 by 10 raised to the power of 2. That is \(5.451 \times 10^2 = 5.451 \times 100\), which results in the decimal number 545.1.

f. 2.934 × 10^(-5)

To convert this number, we will multiply 2.934 by 10 raised to the power of -5. That is \(2.934 \times 10^{-5} = 2.934 \times 0.00001\), which results in the decimal number 0.00002934.

Key concepts

Decimal Conversion

Decimal conversion is a fundamental skill in mathematics that involves changing numbers from one form to another, usually between scientific notation and "ordinary" decimal numbers. When a number is presented in scientific notation, it is expressed as a product of a coefficient and a power of ten, for instance, \(6.244 \times 10^3\). To convert this into a decimal number, you multiply the coefficient by the power of ten it is paired with.
In the example \(6.244 \times 10^3\), multiply 6.244 by 1000, which is the value of \(10^3\). This results in the decimal number 6244. The conversion process simplifies large or small numbers, making them easier to read and understand. Remember, if the power of ten has a negative exponent, as in \(9.117 \times 10^{-2}\), you divide by a power of ten, making the number smaller and resulting in 0.09117.

• Positive exponents make the decimal larger. Move the decimal point to the right.
• Negative exponents make the decimal smaller. Move the decimal point to the left.

Practicing this conversion can help students handle both very large and very small numbers efficiently.

Powers of Ten

Understanding powers of ten is crucial for grasping scientific notation and performing decimal conversions. A power of ten refers to 10 raised to an exponent. This exponent tells you how many times to multiply or divide the number 10.
Consider the basic examples:

• \(10^1 = 10\) is 10, because "10 to the power 1" means 10 multiplied by itself once.
• \(10^2 = 100\) is 100, indicating 10 multiplied by itself twice.
• \(10^{-1} = 0.1\) shows division by 10 once, making the number smaller.

Utilizing powers of ten is especially useful in measuring and expressing large distances or minuscule measurements in science and engineering.
When you encounter a power with a positive exponent, you'll end up moving the decimal point to the right, increasing the number's value. Conversely, a negative exponent means moving the decimal to the left, reducing the value. For example, \(10^3 = 1000\) implies multiplying a number by 1000, while \(10^{-3} = 0.001\) means dividing by 1000. Becoming comfortable with powers of ten will significantly ease your understanding of scientific notation.

Multiplication in Mathematics

Multiplication is one of the basic arithmetic operations in mathematics. It involves combining equal groups, which is essential when working with scientific notations to perform decimal conversions. When multiplying a number by a power of ten, you alter its scale by moving the decimal point.
For instance, in the multiplication \(8.299 \times 10^1\), the number 8.299 is expanded by one place, resulting in 82.99. Simply put, you shift the decimal one place to the right corresponding to \(10^1\). Similarly, for \(1.771 \times 10^{-4}\), shift the decimal place four places to the left, making it 0.0001771.
The concept of multiplication extends into various mathematical operations and concepts, forming a critical part of equations, functions, and problem-solving. It also has properties such as the commutative property (\(a \times b = b \times a\)), which indicates that multiplicands can change places without affecting the product.
Understanding how to multiply efficiently, especially with powers of ten, equips students to tackle more complex mathematical ideas and challenges.