Question

Use a calculator to evaluate each expression. Write your answer in scientific notation. (Round to three decimal places.)(a) \(\left(9.3 \times 10^{6}\right)^{3}\left(6.1 \times 10^{-4}\right)\) (b) \(\frac{\left(2.414 \times 10^{4}\right)^{6}}{\left(1.68 \times 10^{5}\right)^{5}}\)

Short answer

(a) The solution is \(8.026 \times 10^{20}\). (b) The solution is \(1.52 \times 10^{10}\).

Step-by-step solution

Problem (a): Calculate the expression

The first step is to evaluate the expression using the rules of exponents and multiplication. For the expression \((9.3 \times 10^{6})^{3}(6.1 \times 10^{-4})\), raise the \(9.3 \times 10^{6}\) to the power of 3 and then multiply by \(6.1 \times 10^{-4}\). This gives \(8.02557 \times 10^{20}\).

Problem (a): Round the answer

The next step is to round this result to three decimal places. This gives a final answer of \(8.026 \times 10^{20}\).

Problem (b): Calculate the expression

The initial step for the second expression \(\frac{(2.414 \times 10^{4})^{6}}{(1.68 \times 10^{5})^{5}}\) is applying the power to each number in scientific notation, then divide these two terms. This results in \(0.152243 \times 10^{11} \).

Problem (b): Round the answer

The final step is to round the result to three decimal places, which gives a final answer of \(0.152 \times 10^{11}\) or the equivalent \(1.52 \times 10^{10}\).

Key concepts

Exponents

Exponents help us express numbers in a more compact form. They tell us how many times to multiply a number by itself. For example, in the term \(9.3 \times 10^6\), the exponent 6 indicates that 10 is multiplied by itself six times. This results in a large number yet displayed in a manageable size, which is useful in scientific notation for very big or very small numbers. Exponents in parentheses such as \((9.3 \times 10^{6})^3\) mean we need to apply the power of 3 to everything inside. This results in \((9.3^3 \times 10^{18})\), as we multiply the exponent 6 in the base \(10^6\) by 3, resulting in \(10^{18}\).

This rule simplifies complex calculations, as seen in the exercise: First, apply the power to the base number \((9.3^3)\), and then adjust the exponent of the ten (from 6 to 18), ultimately simplifying mathematical expressions greatly.

Multiplication

Multiplication with scientific notation involves multiplying the coefficients and adding the exponents. For instance, the expression \((9.3 \times 10^6)^3 (6.1 \times 10^{-4})\) involves two multiplication operations. First, multiply within the parentheses. We multiply \(9.3^3\) to get a result, and calculate \(10^{18}\) due to the exponent rule, then combine it with \(6.1 \times 10^{-4}\) by multiplying their coefficients.

When multiplying numbers like these, multiply the terms in front (the coefficients) first, then adjust the powers of ten by adding their exponents: \(10^{18-4}\). This basic multiplication rule is integral when simplifying scientific notation.

Division

Scientific notation division involves dividing the coefficients and subtracting the exponents. Consider the expression \(\frac{(2.414 \times 10^4)^6}{(1.68 \times 10^5)^5}\). First, evaluate the top and bottom separately: raise 2.414 to the power of 6 and 1.68 to 5, before dividing these results.

Next, divide the powers of ten by subtracting the exponents (since laws of exponents state \(a^m/a^n = a^{m-n}\)). We calculate \((4 \times 6) - (5 \times 5) = 24 - 25\), resulting in \(\frac{\text{numerator}}{\text{denominator}} \times 10^{-1}\). This step shows division similar to basic arithmetic but implemented through exponent laws.

Rounding Numbers

Rounding numbers in scientific notation refers to adjusting the coefficient to a specific decimal place for simplicity, such as in our exercise where results are rounded to three decimal places. After calculating, it's important that the final scientific value is simple yet accurate to the significant figures needed.

For instance, with \(8.02557 \times 10^{20}\): rounding this figure to three decimal places results in \(8.026 \times 10^{20}\), ensuring precision yet readability. Rounding follows general math rules: if the next number is 5 or more, you round up. This keeps scientific calculations concise without losing essential accuracy needed in precise contexts.