Question

Multiply without using a calculator. Give your answer in scientific notation. $$\left(2 \times 10^{4}\right)(30,000)$$

Short answer

\(6 \times 10^{8}\)

Step-by-step solution

Express both numbers in scientific notation

The first number is already given in scientific notation as \(2 \times 10^{4}\). To express the second number (30,000) in scientific notation, we need to rewrite it as \(3 \times 10^{4}\).

Multiply the coefficients

Multiply the coefficients (the numbers before the exponents) together. In this case, multiply 2 by 3, which equals 6.

Add the exponents

When multiplying powers of 10, add the exponents together. The sum of the exponents 4 and 4 is 8.

Write the final answer in scientific notation

Combine the result from steps 2 and 3 to express the final answer in scientific notation: \(6 \times 10^{8}\).

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used in sciences to handle vast differences in scale. For instance, the mass of the Earth is about 5,970,000,000,000,000,000,000,000 kilograms, which is cumbersome to read and write. In scientific notation, this would be expressed as \(5.97 \times 10^{24}\) kilograms, which is far more manageable.

In the format \(a \times 10^{n}\), \(a\) is known as the coefficient and must be greater than or equal to 1 and less than 10, and \(n\) is the exponent or power of 10 that tells us how many places the decimal point has been moved. Positive exponents indicate large numbers, while negative exponents indicate small numbers, such as fractions or decimals.

Exponent Operations

Understanding exponent operations is crucial for manipulating numbers in scientific notation efficiently. When multiplying numbers with the same base, you simply add the exponents. For example, \(10^{3} \times 10^{2} = 10^{3+2} = 10^{5}\). When dividing numbers with the same base, you subtract the exponents from each other, such as \(10^{3} \div 10^{2} = 10^{3-2} = 10^{1}\).

Here's a handy list of the basic rules for exponents:

• Multiplying with the same base: add the exponents
• Dividing with the same base: subtract the exponents
• Raising a power to a power: multiply the exponents
• Multiplying different bases with the same exponent: distribute the exponent

Remember, these rules only apply when the base number (the number being raised to a power) is the same. Following these rules allows us to simplify complex expressions and is particularly helpful when working with scientific notation.

Coefficient Multiplication

When multiplying numbers in scientific notation, it's important to first multiply the coefficients before handling the exponents. Coefficients are the numbers that appear before the exponential part of scientific notation. The coefficient multiplication is straightforward: just multiply the numbers as you would with any other non-decimal numbers.

For instance, if you have \(2 \times 10^{4}\) and \(3 \times 10^{4}\), you multiply the 2 and 3 together to get a single coefficient. In this case, 2 multiplied by 3 equals 6. It's essential to keep the numbers within the range of 1 to 10 for the result to be correctly formatted in scientific notation. If the multiplication of coefficients results in a number greater than 10, you would need to adjust the coefficient and increment the exponent accordingly.

Note that if the multiplication of coefficients results in a number less than 1, the scientific notation needs to be adjusted so that the coefficient is again between 1 and 10, and the exponent is decreased accordingly. The precision of coefficients can also affect the accuracy of the final result, and it is something to be mindful of in scientific computations.