Problem 60 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. \(56.29 \times 10^{-30}\)

Step-by-Step Solution

Answer

The scientific notation is \(5.629 \times 10^{-29}\).

Step 1: Understand the Problem

The task is to convert the number \(56.29 \times 10^{-30}\) into scientific notation. Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It consists of a number between 1 and 10 multiplied by a power of 10.

Step 2: Convert the Number Prefix

Identify the portion of the number before the multiplication by a power of ten, which is 56.29 in this case. We need to express this number as a number between 1 and 10. To do this, move the decimal point one place to the left, which gives us 5.629.

Step 3: Adjust the Exponent

Since we moved the decimal point of 56.29 one place to the left to make it 5.629, we must increase the exponent by 1 to account for this shift. The original exponent was \(-30\), and by increasing it by 1, it becomes \(-29\).

Step 4: Write in Scientific Notation

Combine the newly adjusted number and the adjusted exponent to write the number in scientific notation. Therefore, the scientific notation of the original number is \(5.629 \times 10^{-29}\).

Key Concepts

Understanding Powers of Ten

The concept of 'powers of ten' is fundamental to scientific notation. It's a method used to express numbers using the base of 10 raised to a certain power or exponent. This comes in handy when dealing with very large or very small numbers.
When a number is expressed as a power of ten, it has the form:

• 10 raised to a positive exponent indicates a large number, such as 10^3 = 1000.
• 10 raised to a negative exponent indicates a small number, such as 10^{-3} = 0.001.

The exponent indicates how many places the decimal needs to move:

• A positive exponent moves the decimal right, increasing the number's size.
• A negative exponent moves the decimal left, decreasing the number's size.

For instance, in the example 56.29 × 10^{-30}, the power of ten tells us that the decimal point in any resulting number will be shifted 30 places to the left.

The Role of Decimal Notation

Decimal notation is the standard way of writing numbers using a decimal point to separate the whole number from the fractional part. This system, based on powers of ten, allows for a clear representation of a wide range of values, from very large to very small.
Decimal notation is crucial in converting a number into scientific notation. You identify the significant figures in a number and adjust the placement of the decimal point so that it falls immediately after the first non-zero digit.
In the instance of 56.29, shifting the decimal point one place to the left gives us 5.629. This move helps in reformatting the number into a manageable and standard form in scientific notation, ensuring that the coefficient remains between 1 and 10 while adjusting the power of ten accordingly.

Understanding Exponents in Scientific Notation

An exponent is a small number written above and to the right of a base number—it shows how many times the base is multiplied by itself. In scientific notation, exponents are used to express large or small numbers efficiently. The number is expressed as a coefficient (a number between 1 and 10) multiplied by 10 raised to an exponent (showing the number of decimal places).

• Exponents can be positive or negative.
• A positive exponent indicates a shift of the decimal to the right, making the original number larger.
• A negative exponent indicates a shift of the decimal to the left, making the original number smaller.

In the given problem, the exponent -30 is increased to -29 when rewriting 56.29 as 5.629 in scientific notation. This adjustment is necessary to account for moving the decimal point one place to create a coefficient between 1 and 10. Thus, modifying exponents is a critical step in correctly applying scientific notation.