Question

Multiply without using a calculator. Give your answer in scientific notation. $$\left(5 \times 10^{4}\right)\left(8 \times 10^{-3}\right)$$

Short answer

The product is 40 \times 10^{1} or 4 \times 10^{2}.

Step-by-step solution

Multiply the coefficients

Multiply the numerical coefficients that are not part of the exponential expressions. In this case, multiply 5 and 8.

Add the exponents

When multiplying exponential terms with the same base, you add the exponents. In this case, add the exponents of 10, which are 4 and -3.

Write in scientific notation

Combine the product of the coefficients from Step 1 with the exponent found in Step 2 to express the final answer in scientific notation.

Key concepts

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It simplifies numbers by condensing them into two parts: a coefficient and a base raised to an exponent. The coefficient is typically a number between 1 and 10, and the base is usually 10. For example, the number 650 could be written as \(6.5 \times 10^2\) in scientific notation. This method is singularly helpful in fields such as science and engineering where dealing with extreme values is common.

To write a number in scientific notation, you first move the decimal point to create a new number from 1 to 10 (this new number is the coefficient). Then, count the number of places you moved the decimal point and use that number as the exponent. If the original number is greater than 10 (as in our example 650), the exponent will be positive. If the number is less than 1, the exponent will be negative.

Exponent Rules

Exponent rules are fundamental laws that help us handle mathematical expressions involving powers efficiently. When numbers with the same base are multiplied, the exponents can be added to simplify the expression. For example, \(10^4 \times 10^{-3}\) becomes \(10^{4 + (-3)}\) which is \(10^1\).

It's essential to remember that exponent addition only applies when the bases are the same. Other crucial exponent rules include:

• When exponents with the same base are divided, subtract the exponents.
• Raising a power to another power means multiplying the exponents.
• A number raised to the power of 0 is always 1.
• Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent.

Understanding these rules is important because it allows for simplification of complex algebraic expressions and ensures accurate calculations.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks of mathematics. When applied to scientific notation, it's critical to carry out these operations while paying attention to the unique format of scientific notation. As seen in the exercise, we first multiply the coefficients (5 and 8). Then, we tackle the exponential part by applying our exponent rules (specifically, the rule for multiplying powers with the same base).

For students aiming to improve their skills in this area, consider visualizing the problem step by step and keep track of both parts of the scientific notation separately—first address the coefficients, then the exponents. Using these organized steps, you can easily handle more complex operations without a calculator.