Question

In Problems 129-136, write each number as a decimal. \(4.112 \times 10^{2}\)

Short answer

411.2

Step-by-step solution

Understand Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

Identify the Components

In the given expression, the number is \(4.112\) and the exponent is \(10^2\).

Apply the Exponent

To remove the scientific notation, move the decimal point in \(4.112\) two places to the right because the exponent is positive 2.

Perform the Calculation

Moving the decimal point two places to the right, \(4.112\) becomes \(411.2\).

Key concepts

Decimal Conversion

Decimal conversion is the process of transforming a number written in scientific or another non-decimal format into a standard decimal number.
In this example, we have the number in scientific notation: \(4.112 \times 10^2 \).
To convert the number from scientific to decimal, we need to deal with the power of ten.
Here, the exponent is 2, meaning we move the decimal point in 4.112 two places to the right. This converts the number to 411.2.
You've effectively translated the notation into a familiar decimal form.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent.
In the format \(a^b\), 'a' is the base, and 'b' is the exponent.
The exponent tells you how many times to multiply the base by itself.
For example, \(10^2\) means 10 multiplied by itself twice, resulting in 100.
In our original problem, the exponent tells us to multiply 4.112 by 100. Hence, \(4.112 \times 10^2 = 4.112 \times 100 = 411.2\).
This clears the basis of applying exponentiation in decimal conversion.

Powers of Ten

Powers of ten are an essential concept in scientific notation.
A power of ten is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times.
Positive powers of ten (\(10^1, 10^2, 10^3, etc.\)) move the decimal point to the right.
Conversely, negative powers of ten (\(10^{-1}, 10^{-2}, etc.\)) move the decimal point to the left.
In our exercise, \(10^2\) moved the decimal in 4.112 two places right, turning it into 411.2.
This fundamental helps bridge the gap in understanding scientific notation and its translation to decimals.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
It's the underpinning for various concepts in mathematics, including scientific notation and exponentiation.
For our problem, we can think of \(4.112 \times 10^2\) as a simple algebraic expression.
By applying the rules of exponentiation and properties of powers of ten, we simplify the expression to get the decimal form 411.2.
Understanding how algebra integrates with these concepts ensures a more comprehensive grasp of scientific notation and its practical application.