Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(4 \times 10^{-2}\right) \cdot\left(3 \times 10^{6}\right) $$

Short answer

The expression evaluates to \(1.2 \times 10^{5}\) in scientific notation and 120000 in decimal form.

Step-by-step solution

Understand scientific notation

In scientific notation, a number is written as a product of two parts: a coefficient and 10 raised to an exponent. Here, \(4 \times 10^{-2}\) and \(3 \times 10^{6}\) are in scientific notation, where 4 and 3 are the coefficients and -2 and 6 are the exponents of 10, respectively.

Multiply coefficients and add exponents

According to the rule of multiplication in scientific notation, coefficients are multiplied together and exponents are added. So, multiply 4 and 3 to get 12, and add -2 and 6 to get 4.

Write the result in scientific notation and decimal form

The result in scientific notation is \(12 \times 10^{4}\). However, in proper scientific notation, the coefficient should be between 1 and 10. So, move the decimal point in 12 one place to the left to get 1.2 and increase the exponent by 1 to get 5. Therefore, the answer in scientific notation is \(1.2 \times 10^{5}\). To write this in decimal form, move the decimal point in 1.2 five places to the right, resulting in 120000.

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or small to be conveniently written in decimal form. It involves two parts: a coefficient and a base of 10 raised to an exponent. The coefficient is a number greater than or equal to 1 and less than 10, and the exponent is an integer. For example, the expression \(4 \times 10^{-2}\) has a coefficient of 4 and an exponent of -2.

When converting to scientific notation, you should make sure the coefficient fits these criteria. If the coefficient is 10 or larger, you'll need to adjust it by moving the decimal point to the left and increasing the exponent by 1 for each move. Conversely, if the coefficient is less than 1, you would move the decimal point to the right and decrease the exponent correspondingly. Understanding scientific notation is crucial for working with large or small numbers in many fields of science and engineering.

Multiplication of Scientific Notations

When multiplying numbers in scientific notation, it's important to remember that you are combining two parts: the coefficients and the powers of ten. The process can be broken down into two key steps:

First, multiply the coefficients together. Second, add the exponents of the base 10. For example, given \(4 \times 10^{-2}\) and \(3 \times 10^{6}\), you multiply the 4 and 3 to get 12 for the coefficients part. Then you add the exponents -2 and 6 to arrive at 4. The multiplication of scientific notations simplifies the handling of very large or very small numbers by turning multiplication into easier operations, such as simple addition of exponents and basic multiplication of small numbers.

Decimal Form Conversion

Converting from scientific notation to decimal form involves moving the decimal point of the coefficient to the right or left, depending on the exponent's sign and magnitude. For a positive exponent, move the decimal point to the right – each move represents a power of ten. For a negative exponent, move it to the left. For instance, \(1.2 \times 10^{5}\) in scientific notation is equivalent to 120,000 in decimal form because the decimal point in 1.2 is moved five places to the right, filling with zeros as needed. Through this conversion, scientific notation can be easily translated into a standard numerical form that can be more intuitively understood and compared.

Exponent Arithmetic

When performing operations with exponents during multiplication and division, there are specific rules to follow. In multiplication, like in our example with \(4 \times 10^{-2}\) and \(3 \times 10^{6}\), you add the exponents when the bases are the same. Thus the resulting exponent is 4, from -2 + 6. It's important to remember that these rules are shortcuts for what's actually happening: you're multiplying or dividing powers of ten, and exponents are simply a shorthand way to express how many times the base number is being multiplied or divided. Mastering exponent arithmetic is essential as it comes up frequently in scientific calculations and helps with simplifying complex expressions.