Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \frac{1.4 \times 10^{-1}}{3.5 \times 10^{-4}} $$

Short answer

The result in scientific notation is \(4.0 \times 10^2\) and in decimal form is 400.0.

Step-by-step solution

Divide The Coefficients And Subtract The Powers of Ten

To start the division, we divide the coefficients (1.4 over 3.5) and subtract the exponent of the denominator from the exponent of the numerator in the powers of ten (-1 minus -4). This is the proper application of exponent arithmetic when dividing terms. Therefore, we get: \[ \frac{1.4 } {3.5} \times 10^{-1 - ( -4)} \]

Simplify The Coefficient And The Exponential Part

Firstly, we simplify the coefficients 1.4/3.5 to get 0.4 and simplify the exponential part to get \(10^3\). So the expression becomes: \[ 0.4 \times 10^3 \].

Adjust the Result to Scientific Notation

To convert to proper scientific notation, we recall that there must be one nonzero digit to the left of the decimal point. Therefore, we adjust the expression by one decimal place to the right: \[ 4.0 \times 10^2 \].

Conversion to Decimal Form

To convert to decimal form, we remember that \(10^2\) means we move the decimal point of the coefficient 2 places to the right which gives us: \[ 400.0 \].

Key concepts

Scientific Notation

Scientific notation is a way to express very large or small numbers in a concise, standardized format. The format typically involves a coefficient – a number between 1 and 10 – followed by a multiplication sign and a power of ten. For example, the number 500 can be written in scientific notation as \(5 \times 10^2\).

This not only helps in simplifying the process of dealing with cumbersome numbers but also makes calculations like multiplication and division more manageable. An essential aspect of scientific notation is placing a single non-zero digit to the left of the decimal point, which was demonstrated in the exercise where we adjusted \(0.4 \times 10^3\) to \(4.0 \times 10^2\) for proper notation.

Exponent Arithmetic

Exponent arithmetic is crucial when working with powers of ten, especially in scientific notation. When dividing numbers in scientific notation, you subtract the exponent in the denominator from the exponent in the numerator. This rule helps simplify the operations without having to compute the powers of ten directly.

In the given exercise, the expression \(10^{-1} / 10^{-4}\) becomes \(10^{-1 - (-4)}\) which simplifies to \(10^3\), because subtracting a negative is equivalent to adding a positive. Understanding this concept is vital for quickly evaluating expressions without a calculator, turning a potentially complex division into an elementary subtraction problem.

Division of Coefficients

In scientific notation, before dealing with the powers of ten, it's necessary to handle the division of coefficients. These are the numbers in front of the multiplication sign. When you have a division like \(1.4 / 3.5\), you simply perform the division as you would with any other numbers.

In the provided exercise, we divided 1.4 by 3.5 to get 0.4. This step is separate from processing the powers of ten but is equally important as it determines the final coefficient for the scientific notation. It's essential to simplify the coefficients as much as possible - they should be reduced to a form that fits within the conventions of scientific notation, between 1 and 10.