Question

For each of the following numbers, if the number is rewritten in standard scientific notation, will the exponent of the power of 10 be positive or negative? a. 1,942,200 b. 15 c. 0.151 d. 0.0000000721

Short answer

For each of the numbers rewritten in standard scientific notation: a. The exponent is positive (\(1.9422 \times 10^6\)). b. The exponent is positive (\(1.5 \times 10^1\)). c. The exponent is negative (\(1.51 \times 10^{-1}\)). d. The exponent is negative (\(7.21 \times 10^{-8}\)).

Step-by-step solution

a. Rewrite 1,942,200 in standard scientific notation

To rewrite the number 1,942,200 in standard scientific notation, we need to find the value of a and n such that the given number can be represented as \(a \times 10^n\). To do this, place the decimal point after the first non-zero digit and then count the number of places the decimal was moved. So, 1,942,200 becomes 1.9422. The decimal point was moved 6 places to the left. Therefore, the number in standard scientific notation is \(1.9422 \times 10^6\). Since n is positive, the exponent is positive.

b. Rewrite 15 in standard scientific notation

To rewrite the number 15 in standard scientific notation, we need to find the value of a and n such that the given number can be represented as \(a \times 10^n\). To do this, place the decimal point after the first non-zero digit and then count the number of places the decimal was moved. In this case, 15 becomes 1.5. The decimal point was moved 1 place to the left. Therefore, the number in standard scientific notation is \(1.5 \times 10^1\). Since n is positive, the exponent is positive.

c. Rewrite 0.151 in standard scientific notation

To rewrite the number 0.151 in standard scientific notation, we need to find the value of a and n such that the given number can be represented as \(a \times 10^n\). To do this, move the decimal point to the spot after the first non-zero digit and count the number of places the decimal was moved. So, 0.151 becomes 1.51. The decimal point was moved 1 place to the right. Therefore, the number in standard scientific notation is \(1.51 \times 10^{-1}\). Since n is negative, the exponent is negative.

d. Rewrite 0.0000000721 in standard scientific notation

To rewrite the number 0.0000000721 in standard scientific notation, we need to find the value of a and n such that the given number can be represented as \(a \times 10^n\). To do this, move the decimal point to the spot after the first non-zero digit and count the number of places the decimal was moved. So, 0.0000000721 becomes 7.21. The decimal point was moved 8 places to the right. Therefore, the number in standard scientific notation is \(7.21 \times 10^{-8}\). Since n is negative, the exponent is negative.

Key concepts

Exponents in Scientific Notation

Understanding exponents in scientific notation is essential for dealing with large or small numbers efficiently. In scientific notation, a number is written as the product of two factors: a coefficient that is at least 1 but less than 10, and a power of 10. This power of 10 is expressed with an exponent, which indicates how many times 10 is to be multiplied by itself.

For instance, the number 600 is written as \(6 \times 10^2\) in scientific notation. The '2' here represents the exponent, showing that the decimal point has moved two places to the right from its original position in the number 6. The power of 10 effectively 'scales' the number up or down, making it easier to work with values that are extremely large or extremely small.

Distinguishing whether the exponent is positive or negative is key. A positive exponent, as seen with \(1.9422 \times 10^6\), shows that we have a large number, where the decimal has moved to the left. Conversely, a negative exponent, like \(7.21 \times 10^{-8}\), signifies a small number, with the decimal moving to the right. Grasping this concept allows students to easily transition between standard form and scientific notation, a skill that's critical in many scientific and mathematical contexts.

Writing Numbers in Scientific Notation

When writing numbers in scientific notation, there's a simple process to follow. The goal is to express the number with one non-zero digit to the left of the decimal point, followed by the remaining significant figures, and then multiply by a power of 10 with the appropriate exponent.

First, identify the most significant digit (the first non-zero digit when looking from left to right for a large number or from right to left for a small number). For a large number like 1,942,200, this digit is '1'. For a small number such as 0.0000000721, it's '7'. Next, place the decimal after this digit and write all other significant digits following it. Finally, determine the exponent for the power of 10 by counting the number of places the decimal has been moved from its original position to reach the new format. If you move the decimal to the left, the exponent is positive; if you move it to the right, it's negative. For example, 15 is written as \(1.5 \times 10^1\) because the decimal moved one place to the left, resulting in a positive exponent.

By consistently applying this process, students can confidently convert any standard number into its scientific notation form which can simplify calculations and comparisons of vastly different values.

Positive and Negative Exponents

Positive and negative exponents in scientific notation have specific meanings that relate to the magnitude of the number. A positive exponent, found by moving the decimal to the left, signifies a number greater than 1. It's a way to efficiently write large numbers without writing out a long string of zeros. For example, \(3.4 \times 10^3\) represents 3400, showing that the number is in the thousands.

On the other hand, a negative exponent results from moving the decimal to the right and denotes a number less than 1, often a fraction or a decimal. It simplifies the expression of very small numbers. For example, \(5.67 \times 10^{-4}\) indicates 0.000567. Negative exponents reflect the division by a power of 10, thus reducing the number's size.

Remember, the exponent merely shows the number of places the decimal has moved, not the value of the number itself. The direction of movement determines the sign of the exponent. Recognizing this concept is crucial for interpreting and manipulating scientific data, as it often involves comparing quantities of various scales and sizes.