Question

For each of the following numbers, if the number is rewritten in scientific notation, will the exponent of the power of 10 be positive, negative, or zero? a. \(1 / 10^{3}\) b. 0.00045 c. 52,550 d. 7.21 e. \(1 / 3\)

Short answer

a. Negative b. Negative c. Positive d. Zero e. Zero

Step-by-step solution

(Determine the exponent for a: \(1 / 10^{3}\))

We can rewrite \(1 / 10^{3}\) as \(1 × 10^{-3}\). Here, the exponent is -3, which is negative.

(Determine the exponent for b: 0.00045)

To rewrite 0.00045 in scientific notation, we need to move the decimal point 4 places to the right: \(4.5 × 10^{-4}\). Here, the exponent is -4, which is negative.

(Determine the exponent for c: 52,550)

To rewrite 52,550 in scientific notation, we need to move the decimal point 4 places to the left: \(5.255 × 10^4\). Here, the exponent is 4, which is positive.

(Determine the exponent for d: 7.21)

This number is already in scientific notation: \(7.21 × 10^0\), and the exponent is 0, which is zero.

(Determine the exponent for e: \(1 / 3\))

We need to determine the ratio \(1/3\) first. As a decimal, \(1/3 = 0.3333...\), to rewrite this in scientific notation, we need to move the decimal point 0 places: \(3.333... × 10^{0}\). Here, the exponent is 0, which is zero. To summarize the results: a. negative b. negative c. positive d. zero e. zero

Key concepts

Positive Exponent

Positive exponents indicate that the base number is multiplied by itself a certain number of times. In scientific notation, a positive exponent signals that the original number is larger than 1. This happens when we shift the decimal point to the left to change the number into a form that is between 1 and 10.

For example, look at the number 52,550. In scientific notation, this becomes \(5.255 \times 10^4\). Here, the exponent is 4, which tells us to multiply 5.255 by 10, four times, moving the decimal point to the right 4 positions to reach the original number. A positive exponent makes it clear that the number is quite large and confirms the multiplication by powers of ten.

To summarize, when rewriting a number larger than 1 in scientific notation:

• The decimal point moves to the left.
• The exponent is positive.
• The original number is being multiplied by 10 raised to the power of the exponent.

Negative Exponent

A negative exponent in scientific notation indicates that the base number is divided by itself multiple times, meaning the original number is less than 1. This happens when you move the decimal point to the right.

Take the number 0.00045 as an example. To express this in scientific notation: move the decimal point 4 places to the right, resulting in \(4.5 \times 10^{-4}\). The negative exponent \(-4\) shows that the number is much smaller than 1, instructing us to divide 4.5 by 10 for four times.

Here's a helpful summary for negative exponents:

• The decimal point moves to the right.
• The exponent is negative.
• The original number is expressed as being divided by 10 raised to the power of the exponent.

Negative exponents are particularly useful for representing tiny numbers while maintaining clarity and brevity, emphasizing how many places the decimal has shifted.

Zero Exponent

In scientific notation, a zero exponent indicates that the base number, after adjusting the decimal, is exactly the number itself. This implies that the number is already in the range of 1 to 10 and does not require moving the decimal point.

For instance, the number 7.21 already meets scientific notation standards with \(7.21 \times 10^0\). The zero exponent shows that 7.21 is already a number with no additional power of ten adjustments needed. In essence, when raised to the power of zero, powers of ten (\(10^0\)) equal one, meaning multiplying by one does not change the number.

Key points for zero exponents:

• The number is already between 1 and 10.
• The exponent is zero, requiring no movement of the decimal point.
• The multiplication by 10 raised to zero results in no change to the number.

This keeps numbers efficiently expressed, highlighting that no further adjustment is necessary to present the number in the desired format.