Problem 22 - Elementary Algebra

Question

Write \(8.90 \times 10^{5}\) in standard form.

Step-by-Step Solution

Answer

Question: Convert the scientific notation \(8.90 \times 10^{5}\) into standard form. Answer: 890,000

Step 1: Understand scientific notation

In scientific notation, a number is represented as the product of two factors: a coefficient (in this case, 8.90) and a power of 10 (in this case, \(10^5\)). The purpose of using this notation is to simplify and easily compare large (or very small) numbers.

Step 2: Manipulate the exponent

To convert the number from scientific notation to standard form, we will need to manipulate the exponent. The exponent represents the number of places the decimal point needs to be moved. In this case, we have \(10^5\), which means we need to move the decimal point 5 places to the right.

Step 3: Convert the number into standard form

Now, we will move the decimal point in the coefficient 8.90, five places to the right as dictated by the exponent: Initial number: 8.90 Step 1: 89.0 (1 place to the right) Step 2: 890 (2 places to the right) Step 3: 8900 (3 places to the right) Step 4: 89000 (4 places to the right) Step 5: 890000 (5 places to the right) After moving the decimal point 5 positions to the right, the number in standard form is 890,000.

Key Concepts

Exponents

An exponent is a mathematical notation indicating the number of times a base number is multiplied by itself. In the expression \( 10^5 \), the number 10 is the base, and 5 is the exponent. This tells us that we need to multiply 10 by itself 5 times, so \( 10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000 \). This concept is crucial in scientific notation because it helps to manage the scale of numbers efficiently.

When working with exponents, it's useful to remember:

• Any number raised to the power of zero is 1.
• An exponent of 1 means the number is unchanged, such as \( 10^1 = 10 \).
• Exponential expressions can be used to represent both large and small numbers more conveniently.

In scientific notation, exponents make it possible to write very large numbers compactly, as with \( 8.90 \times 10^5 \). The exponent here indicates how many places to move the decimal, thereby expanding our number's reach without adding more digits.

Standard Form Conversion

Conversion to standard form helps to translate the compact scientific notation into a conventional numerical format. This involves repositioning the decimal point in accordance with the exponent's direction and magnitude.

Here's how you do it for \( 8.90 \times 10^5 \):

• Identify the coefficient, here 8.90, and the exponent, which is 5.
• The exponent indicates the decimal point should be moved 5 places to the right, transforming 8.90 into a larger whole number.
• Perform this repositioning step by step, ensuring the calculation is precise.

This precise repositioning allows one to convert \( 8.90 \times 10^5 \) into 890,000, thereby making the number legible in standard format. Understanding this conversion is vital as it solidifies one's grasp of how scientific notation simplifies complex calculations in mathematics and science.

Decimal Point Manipulation

Manipulating the decimal point is a key skill when converting between scientific notation and standard form. Shifting the decimal point changes the number's value based on the exponent's instructions.

Let's break down the process using our example \( 8.90 \times 10^5 \):

• The exponent 5 dictates that the point should move 5 places to the right.
• Begin with 8.90 and move the decimal sequentially past each zero to the new position, creating the numbers: 89.0, 890.0, 8900, 89000, and finally 890000.
• Each shift represents one magnitude of 10, exponentially increasing the value.

Remember:

• Moving the decimal point to the right increases the number's value exponentially.
• Conversely, moving it to the left does the reverse, shrinking the number's size.
• This method ensures clarity in interpreting various formats, aiding in accurate math and scientific calculations.

By mastering decimal point manipulation, you're equipped to handle a wide range of numerical challenges across different number formats.