Problem 85
Question
Write each number in scientific notation. $$ -0.00000000504 $$
Step-by-Step Solution
Verified Answer
\(-5.04 \times 10^{-9}\)
1Step 1: Identify starting position
We have a small negative number -0.00000000504. The decimal point is currently located at the left end of the number.
2Step 2: Move the decimal point
We need to obtain a coefficient between 1 and 10, so we move the decimal point towards the right. We land on -5.04 after 9 steps.
3Step 3: Expressing in scientific notation
Putting the coefficient and the power of 10 together, the number in scientific notation is \(-5.04 \times 10^{-9}\).
Key Concepts
Expressing Numbers in Scientific NotationCoefficient in Scientific NotationNegative Exponents
Expressing Numbers in Scientific Notation
Scientific notation is a method of writing very large or very small numbers in a concise and standardized format. It is especially useful in the fields of science and engineering where such numbers frequently occur. To express a number in scientific notation, you first determine the coefficient - a number between 1 and 10 - and then multiply it by a power of 10.
In the provided exercise example, the number \( -0.00000000504 \) needs to be expressed in scientific notation. This is done by identifying the first nonzero digit in the number and moving the decimal place right next to it, ensuring that the coefficient remains between 1 and 10. Once you have obtained the coefficient, you count the number of places the decimal has moved. This count will become the exponent of 10 in the scientific notation.
When dealing with small numbers, like in the example, the decimal point is moved to the right and the exponent given to the power of 10 will be negative. The process allows us to rewrite \( -0.00000000504 \) as a product of the coefficient \( -5.04 \) and the power of 10, resulting in the scientific notation \( -5.04 \times 10^{-9} \).
In the provided exercise example, the number \( -0.00000000504 \) needs to be expressed in scientific notation. This is done by identifying the first nonzero digit in the number and moving the decimal place right next to it, ensuring that the coefficient remains between 1 and 10. Once you have obtained the coefficient, you count the number of places the decimal has moved. This count will become the exponent of 10 in the scientific notation.
When dealing with small numbers, like in the example, the decimal point is moved to the right and the exponent given to the power of 10 will be negative. The process allows us to rewrite \( -0.00000000504 \) as a product of the coefficient \( -5.04 \) and the power of 10, resulting in the scientific notation \( -5.04 \times 10^{-9} \).
Coefficient in Scientific Notation
The coefficient is a crucial part of scientific notation. It is the term that carries the significant figures of the original number and remains between 1 and 10. This constraint ensures that scientific notation is consistent and easily comparable across different numbers.
For example, in scientific notation, the coefficient of \( -5.04 \times 10^{-9} \) is \( -5.04 \). This number is what was left after the decimal point was adjusted during the conversion process. The significance of having a coefficient within this range is to prevent additional zeros in the coefficient itself, thus maintaining the simplicity of the expression.
Moreover, the sign of the coefficient (- in this case) indicates if the original number was positive or negative, which is an important feature of scientific notation as it preserves the number's authenticity.
For example, in scientific notation, the coefficient of \( -5.04 \times 10^{-9} \) is \( -5.04 \). This number is what was left after the decimal point was adjusted during the conversion process. The significance of having a coefficient within this range is to prevent additional zeros in the coefficient itself, thus maintaining the simplicity of the expression.
Moreover, the sign of the coefficient (- in this case) indicates if the original number was positive or negative, which is an important feature of scientific notation as it preserves the number's authenticity.
Negative Exponents
Negative exponents in scientific notation represent how small a number is and the direction the decimal point has moved to get the coefficient. In scientific notation, a negative exponent tells us that we are dealing with a fraction, specifically a number that is less than one.
To understand the negative exponent, consider the number \( 10^{-9} \). This is equivalent to \( \frac{1}{10^9} \), meaning one divided by ten raised to the ninth power. So the negative exponent shows how many times you would divide by 10 to achieve the original number.
In our example, the exponent is -9, which shows that the decimal point moved 9 places to the right to get from 0.00000000504 to 5.04. Therefore, \( -5.04 \times 10^{-9} \) tells us that the original number is 5.04 divided by ten billion (10^9). Negative exponents are essential for writing extremely small numbers succinctly in scientific notation.
To understand the negative exponent, consider the number \( 10^{-9} \). This is equivalent to \( \frac{1}{10^9} \), meaning one divided by ten raised to the ninth power. So the negative exponent shows how many times you would divide by 10 to achieve the original number.
In our example, the exponent is -9, which shows that the decimal point moved 9 places to the right to get from 0.00000000504 to 5.04. Therefore, \( -5.04 \times 10^{-9} \) tells us that the original number is 5.04 divided by ten billion (10^9). Negative exponents are essential for writing extremely small numbers succinctly in scientific notation.
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Problem 85
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