Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate each expression. Write your answer in scientific notation. (Round to three decimal places.)(a) \(\left(1.2 \times 10^{2}\right)^{2}\left(5.3 \times 10^{-5}\right)\) (b) \(\frac{\left(3.28 \times 10^{-6}\right)^{10}}{\left(5.34 \times 10^{-3}\right)^{22}}\)

Step-by-Step Solution

Answer

(a) The solution is approximately \(7.632 \times 10^{-1}\). (b) The solution is around \(2.167 \times 10^{7}\)

Step 1: Problem (a): Square and Multiply

Begin by squaring the \(1.2 \times 10^{2}\). The rule for squaring numbers in scientific notation is to square the numeric part and add the exponents, so it becomes \(1.44 \times 10^{4}\). Then, multiply this result by \(5.3 \times 10^{-5}\). (Remember that when you multiply numbers in scientific notation, you multiply the numeric parts and add the exponents.) Finally, you should get approximately \(7.632 \times 10^{-1}\)

Step 2: Problem (b): Raise to Power and Divide

Start with \(3.28 \times 10^{-6}\) raised to the 10th power, which becomes approximately \(1.847 \times 10^{-59}\). (Remember that when you raise a number in scientific notation to a certain power, you raise the numeric part to that power, and multiply the exponent by the power.) Then do the same with the denominator, and get approximately \(8.524 \times 10^{-66}\). Finally, divide the two results. (When dividing, you divide the numeric parts and subtract the exponent of the denominator from that of the numerator.) Your final answer should be around \(2.167 \times 10^{7}\).

Key Concepts

Squaring Numbers in Scientific Notation

When it comes to squaring numbers in scientific notation, it involves handling both the coefficient (the numeric part before the exponential part) and the exponent separately. To square a number, you simply square the coefficient and double the exponent.

For example, if we have the number in scientific notation such as \(1.2 \times 10^2\), squaring this number means calculating \((1.2)^2\) and \(2 \cdot 2\) for the exponent. The result is \(1.44 \times 10^4\). It’s essential to remember that the exponent is just doubled, which can be neatly understood as adding it to itself, reflecting the nature of squaring as a form of repeated multiplication.

Multiplying in Scientific Notation

Multiplying numbers in scientific notation is a straightforward process. In essence, you multiply the coefficients (the numbers in front of the \(10\)) and then add the exponents. The logic behind this is that you’re multiplying powers of ten, which algebraically translates to adding the exponents.

For example, multiplying \(1.44 \times 10^4\) and \(5.3 \times 10^{-5}\) means doing the following: multiply the coefficients (\(1.44\) and \(5.3\)) to get \(7.632\), and then add the exponents (\(4\) and \(-5\)) to get \(-1\). The product is written as \(7.632 \times 10^{-1}\). It’s important to remember that adding a negative exponent is the same as subtracting that value if it were positive.

Raising to a Power in Scientific Notation

Raising a number in scientific notation to a power means taking the coefficient and exponent and applying the power to them independently. When raising the coefficient, you simply raise it to the given power. For the exponent, you multiply it by the power.

As in our example, \((3.28 \times 10^{-6})^{10}\) requires us to calculate \((3.28)^{10}\) and \(-6 \times 10\). For practical purposes, this operation can usually be done with a calculator, especially for high powers. The result would be a new number in scientific notation reflecting the new exponent and coefficient.

Dividing in Scientific Notation

Dividing in scientific notation is just as intuitive as multiplying. Here, you’ll divide the coefficients and subtract the exponents. Subtracting the exponents is directly related to the rules of dividing powers; when you divide same bases with exponents, you subtract the exponents.

Following the exercise given, when we divide \(1.847 \times 10^{-59}\) by \(8.524 \times 10^{-66}\), we divide the coefficients (\(1.847\) by \(8.524\)) and subtract the exponent of the denominator from the exponent of the numerator (\(-59 - (-66)\)). This yields a final answer in scientific notation that represents the quotient with the correct coefficient and exponent.