Question

Write each number in scientific notation. 32,155

Short answer

3.2155 \times 10^4

Step-by-step solution

Identify the given number

The number provided in the problem is 32,155.

Place the decimal point

Place the decimal point after the first non-zero digit. For 32,155, this would be 3.2155.

Determine the exponent

Count the number of places the decimal point has moved from its original position to its new position. In this case, it moved 4 places to the left.

Write in scientific notation

Combine the decimal point positioning and the exponent. The result is \[3.2155 \times 10^4\].

Key concepts

Decimal Point

The decimal point is a crucial component in scientific notation. It separates the integer part of a number from its fractional part. For example, in the number 32,155, the decimal point initially appears at the end (implied) of the number. To convert this into scientific notation, first, you need to move the decimal point to the position after the first non-zero digit. This means 32,155 becomes 3.2155. Placing the decimal at this position helps in expressing the number in a standard form that is between 1 and 10. This consistent placement makes it easier to handle and compare large or small numbers.

Exponent

The exponent in scientific notation tells you how many times the number is multiplied by 10. In our example, the original number 32,155 is converted to 3.2155. To find the exponent, count how many places the decimal point was moved. Here, it was moved 4 places to the left. So the exponent will be 4. The full scientific notation of the number would then be expressed as ** 3.2155 x 3.2155 x 10^4. . This 10^4. form makes it easier to read and understand large numbers in a compact form. It also becomes more straightforward when performing mathematical operations.

Powers of Ten

Understanding powers of ten is fundamental to mastering scientific notation. Powers of ten are simply the number 10 raised to various exponents. For instance, 10^1 means 10, 10^2 means 100, and 10^4 means 10,000. To represent the number 32,155, you move the decimal point 4 places to the left, changing the number to 3.2155. Consequently, the power of ten associated with this movement is 10^4. This means that 3.2155 is multiplied by 10 raised to the power of 4, or 10,000, which gives back the original number when expanded. This way of expressing numbers allows us to handle very large and very small numbers in a concise, easy-to-manage form. Mastering powers of ten will improve your confidence in handling a variety of mathematical situations, whether in school or professional settings.