Problem 7 - Algebra 1: Concepts and Skills

Question

Write the number in decimal form. $$ 2.45 \times 10^{-1} $$

Step-by-Step Solution

Answer

The number in decimal form is 0.245

Step 1: Understand the notation

The given number is in scientific notation, which is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. The standard format for scientific notation is \(a \times 10^n\) where \(n\) is an integer, and \(a\) is a real number. In this case, \(a = 2.45\) and \(n = -1\).

Step 2: Transpose to decimal form

When the exponent of 10 is a negative number, we move the decimal point in \(a\) to the left by that many places. Here, since our exponent is -1, we move the decimal point in 2.45 one place to the left.

Step 3: Write the number in decimal form

After moving the decimal point one place to the left, the number in decimal form is 0.245

Key Concepts

Scientific Notation

Scientific notation is a method used to write very large or very small numbers in a compact form. It allows us to express these numbers as a product of a decimal and a power of ten. This notation makes it easier to read, write, and understand such numbers.
When using scientific notation, a number is typically written in the form of \(a \times 10^n\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer. This format is used because it's more convenient and can simplify complex calculations.
For example, the number 245,000 can be represented as \(2.45 \times 10^5\), simplifying its representation while still maintaining its value. In the exercise above, the number \(2.45 \times 10^{-1}\) is an example of how scientific notation is utilized for small numbers.

Decimal Point Movement

The movement of the decimal point plays a crucial role in converting a number from scientific notation to decimal form. The exponent in scientific notation dictates how and where the decimal point moves. This is the key step in transitioning a number from one form to the other.

• If the exponent is positive, move the decimal point to the right.
• If the exponent is negative, move the decimal point to the left.

For example, with the number \(2.45 \times 10^{-1}\), the exponent is \(-1\). Thus, you move the decimal one position to the left, changing the number to 0.245. The movement of the decimal point transforms the number so that it can be read naturally and precisely in decimal form.

Exponents in Scientific Notation

Exponents in scientific notation indicate the number of times the base (which is 10) is multiplied by itself. They determine how the decimal point in the coefficient \(a\) should be moved.
The exponent \(n\) shows how many places the decimal point should move and in which direction:

• A positive exponent means you multiply the number by 10 raised to the power of that exponent, moving the decimal to the right.
• A negative exponent means you divide the number by 10 raised to the absolute value of that exponent, moving the decimal to the left.

In the original exercise problem, the exponent is \(-1\), signaling that the decimal point in 2.45 should shift leftward by one spot. This exponent manipulation is essential to accurately converting scientific notation to its equivalent decimal form.