Problem 56 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \left(8 \times 10^{-3}\right)\left(7 \times 10^{1}\right) $$

Step-by-Step Solution

Answer

Scientific notation: \(5.6 \times 10^{-1}\); Standard form: 0.56.

Step 1: Understand Scientific Notation

Scientific notation is a way to write large or small numbers using powers of ten. In this problem, we have two numbers expressed this way: \(8 \times 10^{-3}\) and \(7 \times 10^{1}\).

Step 2: Multiply the Coefficients

Identify the coefficients of the two numbers, which are 8 and 7 in this case. Multiply them together: \(8 \times 7 = 56\).

Step 3: Multiply the Powers of Ten

In scientific notation, when multiplying, you add the exponents of 10. Here, the exponents are -3 and 1. So, add them: \(-3 + 1 = -2\).

Step 4: Combine Results in Scientific Notation

Combine the result from Steps 2 and 3 to form the scientific notation of the answer: \(56 \times 10^{-2}\).

Step 5: Express Result in Standard Form

Convert the scientific notation \(56 \times 10^{-2}\) to standard form by moving the decimal point two places to the left, resulting in 0.56.

Key Concepts

Standard Form

Standard form is another way of expressing numbers, mainly used to avoid dealing with an extensive list of zeros. Instead of writing long numbers in full, we can use a much shorter version called 'standard form.'
For instance, the number 0.56, as seen in the exercise solution, is in its standard form. This means you write the number without any exponent. The conversion of a scientific notation into a standard form involves either shifting the decimal to the left (for negative powers of ten) or to the right (for positive powers of ten).
To convert from scientific notation such as \(56 \times 10^{-2}\) to standard form, move the decimal point the same number of spaces as the exponent indicates. Here the exponent is -2, so we shift the decimal place two positions to the left, resulting in 0.56.
This process highlights the ease of readability that standard form offers while making arithmetic operations more manageable.

Powers of Ten

The concept of powers of ten is fundamental in scientific notation and standard form.
It allows us to easily express extremely large or incredibly small numbers. The base number is 10 and it is raised to a power (exponent) which tells us how many times to multiply the base by itself.

• A positive exponent means the number is larger than one. For example, \(10^{3} = 1000\).
• A negative exponent indicates the number is a fraction smaller than one. For example, \(10^{-3} = 0.001\).

In the context of the given exercise, we dealt with powers of ten represented as \(10^{-3}\) and \(10^{1}\). By adding the exponents (as explained in multiplication of exponents), we simplified the expression to \(10^{-2}\). This showcases the utility of powers of ten in simplifying expressions and making them easier to handle.

Multiplication of Exponents

Multiplying numbers expressed in scientific notation involves working with their exponents.
When you multiply powers with the same base, the key rule is to add the exponents. This is because multiplying powers of ten makes use of the properties of exponents:

• \(a^{m} \times a^{n} = a^{m+n}\)

In the provided exercise, \(8 \times 10^{-3}\) and \(7 \times 10^{1}\) involve multiplication where both numbers are powers of ten raised to different exponents. When these were multiplied, the bases (10) remained the same, and the exponents -3 and 1 were added together to give a new exponent of -2.
Therefore, using this rule, we obtained a merged expression that was easier to compute: \(56 \times 10^{-2}\). This shows how multiplication of exponents can simplify complex numerical expressions in scientific calculations.