Problem 9 - Prealgebra

Question

Express each number in standard form. $$4.24 \times 10^{2}$$

Step-by-Step Solution

Answer

424

Step 1: Understanding Standard Form

The standard form of a number is written as a product of a number (usually between 1 and less than 10) and a power of 10. In this case, the number is already in the form of \(4.24 \times 10^{2}\). This means it is 4.24 multiplied by 10 raised to the power of 2.

Step 2: Apply the Power of 10

The exponent 2 on the 10 indicates that we should move the decimal point in the number 4.24 to the right by 2 places. This is because multiplying by 10 shifts a number to the left of the decimal point.

Step 3: Shift the Decimal Point

Move the decimal point in 4.24 two places to the right: - Originally: 4.24 - Move one place right: 42.4 - Move one more place right: 424

Step 4: Write in Standard Form

After shifting the decimal point two places to the right, the number becomes 424. Therefore, the standard form of \(4.24 \times 10^{2}\) is simply writing the result: 424.

Key Concepts

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers quickly and easily. It involves two main components: a decimal number and a power of ten. The decimal number should be at least 1 but less than 10. For instance, the number 4.24 in the given exercise perfectly fits this criterion as it lies within this range. The power of 10 then serves to scale this decimal number to the actual value. In scientific notation, the general format is:

• A number between 1 and less than 10
• Multiplied by 10 raised to an exponent

Together, this format helps streamline complex calculations and represent numbers in a more manageable way. Whether you're working with the vast distances in space or the minute measurements in biology, scientific notation helps keep calculations clear and concise.

Exploring the Powers of 10

When dealing with powers of 10, the exponent tells you how many times to multiply the number 10 by itself. For example, in the expression given in the exercise, the power of 10 is written as \(10^{2}\). This indicates that 10 is multiplied by itself two times:

• \(10^1 = 10\)
• \(10^2 = 10 \times 10 = 100\)

Understanding the power of 10 is crucial when converting numbers back and forth between scientific notation and their standard form. It determines the shift of the decimal in the number making the expression manageable and practical for use in various fields. Whether you're scaling up or down, exponents help replace lengthy multiplication steps with a simple notation.

Mastering Multiplying Decimals by Powers of 10

Multiplying decimals by powers of 10 primarily involves moving the decimal point. The power of 10 will tell you how many places to move the decimal. If the exponent is positive, like in the case of \(4.24 \times 10^{2}\), you move the decimal to the right by the number of spaces indicated by the exponent. Here's how it's done:

• Start with 4.24
• Move one place to get 42.4
• Move another place to arrive at 424

This approach simplifies multiplication, making it easier to understand and execute. When working with negative exponents, you would move the decimal to the left instead. Mastering this concept not only helps in scientific notation but also makes everyday arithmetic involving decimals much less intimidating.