Question

Perform the following computations. Display your answer in scientific notation. $$\left(9.83 \times 10^{5}\right) \div\left(2.77 \times 10^{3}\right)$$

Short answer

The answer is \(3.55 \times 10^{2}\).

Step-by-step solution

Divide the coefficients

Divide the two numbers in front of the exponential part, which are 9.83 and 2.77, without the powers of ten. The result of this division will be used in the scientific notation of the final answer.

Subtract the exponents

Since you are dividing numbers in scientific notation, you need to subtract the exponent of the divisor (\(10^{3}\)) from the exponent of the dividend (\(10^{5}\)). This follows the law of exponents that states when you divide two powers with the same base, you subtract the exponents.

Express in scientific notation

Combine the result from step 1 with the exponent result from step 2 to express the answer in scientific notation. If the result has more than one digit before the decimal point, adjust it to have only one non-zero digit before the decimal point and correct the exponent accordingly.

Key concepts

Dividing Scientific Notation

When it comes to handling large or extremely small numbers, scientific notation is a valuable tool, especially for divisions. Dividing scientific notation is a concept that integrates division with the simplicity that scientific notation offers. To divide two numbers written in scientific notation, start by dividing their coefficients, generally decimal numbers usually between 1 and 10. Then, handle the exponent part separately by subtracting the exponent of the divisor from the exponent of the dividend.

For example, with \(9.83 \times 10^{5}\) divided by \(2.77 \times 10^{3}\), you would first divide 9.83 by 2.77. This results in approximately 3.549, which becomes the coefficient for the scientific notation of the answer. The simplification of the process into these manageable steps helps ensure that students can easily progress through the computation.

Laws of Exponents

The laws of exponents are the rules that govern the operations on numbers with exponents. When dividing in scientific notation, one law in particular is key: to divide powers with the same base, subtract the exponent of the bottom number from the exponent of the top number. This is an application of the more general rule for exponents, \( a^n \/ a^m = a^{n-m} \).

In the context of our previous example, \(10^{5} \/ 10^{3}\) would simplify to \(10^{5-3} \), which is \(10^{2}\). This forms the newer exponent of the scientific notation after the division of the coefficients. Following this law leads to correct exponent manipulation, ensuring that computation yields the right power of ten in the scientific notation.

Scientific Notation Computation

Performing scientific notation computation requires familiarity with both scientific notation and the laws of exponents. It involves manipulating both the decimal coefficients and their associated powers of ten. After dividing the coefficients and handling the exponents as previously mentioned, it's crucial to ensure the final result is also in proper scientific notation.

If the coefficient is not between 1 and 10, adjust it by moving the decimal point until it is within this range and change the exponent to compensate for the shift. In our exercise, 3.549 is already within the accepted range, so there's no need to adjust. Therefore, the final answer to our example would be \(3.549 \times 10^{2}\), showcasing an accurate application of scientific notation computation.