Question

Combine without using a calculator. Give your answer in scientific notation. $$\left(3.0 \times 10^{4}\right)+\left(2.1 \times 10^{5}\right)$$

Short answer

\(2.4 \times 10^{5}\)

Step-by-step solution

Align the Exponents

Make sure both terms are in scientific notation with the same power of ten. If necessary, adjust one term's coefficient and exponent so they match. In this case, convert the first term from \(3.0 \times 10^{4}\) to \(0.30 \times 10^{5}\) so that both terms have the same exponent, \(10^{5}\).

Combine the Coefficients

Add the coefficients together while keeping the same power of ten. Combine \(0.30\) and \(2.1\) to get \(0.30 + 2.1 = 2.4\).

Express the Sum in Scientific Notation

Write the final result as \(2.4 \times 10^{5}\) to express the sum in correct scientific notation.

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form that makes them easier to work with. This method uses powers of ten to simplify numbers. In scientific notation, a number is written as the product of a coefficient and a power of ten. The coefficient must be a number equal to or greater than 1 and less than 10, and it is followed by the multiplication sign and the base number 10 raised to an exponent. For example, the scientific notation for 650 is \( 6.5 \times 10^{2} \).

When working with scientific notation, it's essential to understand the meaning of the exponent. The exponent indicates how many times the base, which is 10, is to be multiplied by itself. A positive exponent shows that the number is large, whereas a negative exponent indicates a small number, as the base will be divided accordingly.

Exponent Alignment

When adding or subtracting numbers in scientific notation, a crucial step is to align their exponents. It is necessary because we can only operate with numbers that have the same power of ten directly. If the exponents do not match, we adjust one of the coefficients to ensure that the exponents are the same.

To align the exponents, you convert one of the numbers by moving its decimal point, changing both the coefficient and the exponent until both exponents are identical. For instance, to add \(3.0 \times 10^{4}\) and \(2.1 \times 10^{5}\), adjust the first term to \(0.30 \times 10^{5}\) so that both terms have the power of \(10^{5}\). This makes the addition process straightforward because you are now working with like terms.

Coefficients Addition

Once the exponents are aligned, the addition process focuses on the coefficients. With the exponents matching, add the coefficients as you would with any ordinary numbers. Remember, since scientific notation requires the coefficient to be between 1 and 10, if the addition results in a number outside this range, you will need to adjust the coefficient and the exponent again to maintain proper scientific notation.

Let's consider our example where the aligned terms were \(0.30 \times 10^{5}\) and \(2.1 \times 10^{5}\). To add these coefficients, simply calculate \(0.30 + 2.1 = 2.4\). The result is then combined with the common exponent to get \(2.4 \times 10^{5}\), which is the answer in correct scientific notation.