Question

In Problems 129-136, write each number as a decimal. \(9.7 \times 10^{3}\)

Short answer

9700

Step-by-step solution

Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small in a compact form. It is written as the product of a number between 1 and 10 and a power of 10. In this problem, the number is given as 9.7 and the power of ten is 10^3.

Identify the Components

The given number is 9.7, and it is multiplied by 10 raised to the power of 3. This means we need to move the decimal point in 9.7 three places to the right because the exponent is positive.

Move the Decimal Point

Starting with 9.7, move the decimal point three places to the right. This means adding zeroes as needed: 9.7 becomes 97, and then 970, and finally 9700.

Write the Final Answer

After moving the decimal point, the number expressed as a decimal is 9700.

Key concepts

Decimal Conversion

Understanding how to convert scientific notation to a decimal is essential. Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of ten. For example, in the given exercise, we dealt with the number 9.7 multiplied by 10 raised to the power of 3. To convert this into a decimal, identify the number (9.7 in this case) and understand that the exponent (3) informs us about how many places to move the decimal point. Moving the decimal point correctly is key to accurate decimal conversion.

Powers of Ten

The power of ten tells us how many places to move the decimal point. In our exercise, the power is 3, indicating that we need to move the decimal point three places to the right. This exponent concept reduces large calculations to simpler multiplication or division tasks. For larger exponents, you move the decimal point to the right, significantly increasing the number. Conversely, for negative exponents, you move it to the left, making the number smaller.

Moving Decimal Point

Correctly moving the decimal point transforms the scientific notation into an easy-to-read decimal. Start with the number 9.7 and move the decimal point three places to the right (as indicated by the positive exponent 3). You would see this transformation as:

• Move once: 97.
• Add a zero: 970.
• Add another zero: 9700.

Ensuring each move or zero addition retains accuracy in the decimal conversion. Missing a step or an extra zero can lead to incorrect results, so careful progression is crucial.