Question

Convert these to nonscientific notation. (a) \(3.256 \times\) \(10^{-5},(b) 6.03 \times 10^{6}\)

Short answer

(a) The nonscientific notation of \(3.256 \times 10^{-5}\) is \(.00003256\). (b) The nonscientific notation of \(6.03 \times 10^{6}\) is \(6030000\).

Step-by-step solution

Converting \(3.256 \times 10^{-5}\) to nonscientific notation

A negative exponent of 5 on 10 indicates that the decimal point in 3.256 needs to be moved 5 places to the left. When moving to the left, if we run out of digits, we can just add zeros. Therefore, it will be \(.00003256\).

Converting \(6.03 \times 10^{6}\) to nonscientific notation

A positive exponent of 6 on 10 indicates that the decimal point in 6.03 needs to be moved 6 places to the right. We move it two places to the right to get 603, and then we add four more zeros, resulting in \(6030000\).

Key concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are written as a small number, called the power, placed to the upper right of a base number. For example, in scientific notation, the format of the number often involves an exponent of 10. This indicates how many times the decimal point in the base number needs to be shifted. If the exponent is positive, like in the case of \(10^6\), it shows the number of places the decimal point must move to the right. Conversely, a negative exponent, such as \(10^{-5}\), means the decimal point moves to the left.

• Positive Exponents: Shift decimal to the right.
• Negative Exponents: Shift decimal to the left.

Understanding how exponents work is crucial in scientific notation conversion, as they dictate the direction and distance the decimal needs to move.

Decimal Point Movement

Decimal point movement is a fundamental step in converting numbers between scientific and nonscientific notation. It involves shifting the decimal point in the number based on the value and sign of the exponent. In our example, we are converting numbers like \(3.256 \times 10^{-5}\) and \(6.03 \times 10^{6}\).For \(3.256 \times 10^{-5}\), the negative sign tells us to move the decimal point five places to the left. Thus, the number becomes \(.00003256\). Moving to the left requires filling in the empty places with zeros.On the other hand, for \(6.03 \times 10^{6}\), a positive exponent of 6 indicates that we should move the decimal six places to the right, leading to \(6030000\). Here, we add zeros as necessary to occupy the places.Making sure each movement accounts for every decimal position correctly ensures the conversion is accurate every time.

Converting to Nonscientific Notation

Nonscientific notation, often referred to as standard form or plain numbers, is the representation of numbers without exponents. Converting from scientific to nonscientific notation involves removing the \(\times 10^n\) part and reorienting the number accordingly. Let's break down the conversion process:

• Analyze the Exponent: Determine the number of decimal movements required.
• Move the Decimal: Adjust the decimal point based on the sign and value of the exponent, shifting left for negative and right for positive exponents.
• Add Zeros if Needed: Introduce zeros to fill in the gaps where there are no digits, ensuring the numerical value is preserved.

In practice, this means transforming \(3.256 \times 10^{-5}\) into \(.00003256\) and \(6.03 \times 10^{6}\) into \(6030000\). Understanding these steps makes scientific notation less intimidating and more manageable.