Question

EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Write the result in scientific notation and in decimal form. $$ \left(7 \times 10^{-3}\right) \cdot\left(8 \times 10^{-4}\right) $$

Short answer

The expression \((7 \times 10^{-3}) \cdot (8 \times 10^{-4})\) evaluates to \(5.6 \times 10^{-6}\) in scientific notation, and it represents 0.0000056 in decimal form.

Step-by-step solution

Identify the Terms

The expression consists of two terms in scientific notation, which are \(7 \times 10^{-3}\) and \(8 \times 10^{-4}\).

Multiply the Base Number

First, multiply the base numbers, which are \7\) and \(8\). The result is \(7 \times 8 = 56\). Since we need the answer in scientific notation, and the base number should be between 1 and 10, divide 56 by 10 to get 5.6 and adjust the exponent by adding 1.

Multiply the Exponent

Then, to multiply the powers of ten, simply add their exponents. The original exponents are \(-3\) and \(-4\). Adding \(-3\) and \(-4\) gives \(-7\). But remember that we previously added 1 to the exponent when adjusting the base number, so you must add this 1 to the \(-7\) to get \(-6\). Therefore, the result in scientific notation will be \(5.6 \times 10^{-6}\).

Convert to Decimal Form

In decimal form, this means moving the decimal point 6 places to the left (because of the negative exponent) of the number 5.6. Since we have only one digit to the left, we will need to add five zeros. So, the decimal equivalent will be 0.0000056

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 often used by scientists, mathematicians, and engineers to make very large or very small numbers easier to work with. The general format for scientific notation is \( a \times 10^n \), where \( a \) is a coefficient that is greater than or equal to 1 and less than 10, and \( n \) is an integer representing the exponent.

For example, in the expression \( 7 \times 10^{-3} \) from our exercise, \( 7 \) is the coefficient and \( -3 \) is the exponent. This notation indicates that the decimal point in the number \( 7 \) should be moved three places to the left, which would make it \( 0.007 \). If the exponent were positive, we'd move the decimal to the right instead. Scientific notation allows for a more compact and precise representation of values, particularly when dealing with calculations in science and engineering.

Exponents

Exponents, also known as powers, are a shorthand way to indicate how many times to multiply a number by itself. The exponent is placed above and to the right of the base number. For instance, in \( 10^2 \), the base is \( 10 \) and the exponent is \( 2 \), telling us to multiply \( 10 \) by itself once: \( 10 \times 10 = 100 \).

When multiplying numbers with exponents that have the same base, you can simplify the process by adding the exponents together. In our textbook example, we multiply \( 10^{-3} \) by \( 10^{-4} \), so we add the exponents \( -3 \) and \( -4 \) to get a new exponent of \( -7 \). Exponents can also be negative, which means that the base number is being divided instead of multiplied. Negative exponents represent fractions, where \( 10^{-1} \) equals \( 1/10 \), and so on.

Multiplication of Powers of Ten

Multiplying powers of ten follows its own set of rules which stem from the nature of exponents. When you multiply two powers of ten, you only need to add the exponents together to find the answer. This works because \( 10^n \times 10^m = 10^{n+m} \). Therefore, each time you multiply by another power of ten, you are essentially shifting the decimal point in the base number, allowing for quick and easy calculations.

Let's look at the exercise \( 7 \times 10^{-3} \times 8 \times 10^{-4} \). When we multiply the coefficients \(7\) and \(8\), we get \(56\), but then we adjust to keep it within the bounds of scientific notation, giving us \(5.6\) and incrementing our exponent by 1. Then, with the exponents \( -3 \) and \( -4 \) from the original powers of ten, we add them along with the extra 1 from adjusting the base number, ending up with \( 10^{-6} \).

This shows us just how elegant and efficient multiplication of powers of ten can be, turning what could be a complex arithmetic operation into a simple matter of adding or subtracting integers (the exponents).