Question

Express each of the following products as a power of 10 : (a) \(10 \times 10 \times 10\) (b) \(1 / 10 \times 1 / 10 \times 1 / 10\)

Short answer

(a) is \(10^3\) and (b) is \(10^{-3}\).

Step-by-step solution

Identifying Patterns in (a)

To solve part (a) \(10 \times 10 \times 10\), first recognize that multiplying the same number repeatedly involves using powers. In this case, we have three 10s being multiplied together.

Expressing as a Power in (a)

Since there are three 10s being multiplied, this is the same as raising 10 to the power of 3. Therefore, the expression can be written as \(10^3\).

Identifying Patterns in (b)

To solve part (b) \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}\), recognize this as multiplying fractions, where each fraction is \(10^{-1}\).

Expressing as a Power in (b)

When multiplying powers of the same base, you add the exponents. Here, add the exponents of \(10^{-1}\) for each term: \(-1 + -1 + -1 = -3\). Thus, the expression is \(10^{-3}\).

Key concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. They give us a concise method to represent these expressions. The number being multiplied is called the "base," and the exponent tells us how many times the base is used as a factor. For instance, in the expression \(10^3\), 10 is the base and 3 is the exponent. This indicates we multiply 10 three times: \(10 \times 10 \times 10\). Using exponents makes calculations simpler and expressions more compact. Whenever you see a number raised to an exponent, remember it signifies repeated multiplication. Adjusting and working with exponents becomes an essential skill, especially when dealing with powers of ten.

Multiplying Powers with the Same Base

When you multiply powers that have the same base, there is a simple rule to follow: add the exponents. This is because each exponent represents the number of times the base is used as a factor. For instance, if you have \(10^2 \times 10^3\), the result is \(10^{2+3} = 10^5\). Why does this work? Consider that \(10^2\) means \(10 \times 10\) and \(10^3\) means \(10 \times 10 \times 10\). If you multiply them together, you end up with \(10 \times 10 \times 10 \times 10 \times 10\), or \(10^5\). Understanding and using this rule can greatly simplify multiplication tasks, particularly when handling large numbers or repeated multiplication.

Fraction Multiplication with Powers

Fraction multiplication can seem tricky, but it becomes manageable when seen through the lens of exponents. Especially when working with fractions that involve powers of ten, it helps to convert them into exponents. For example, \(\frac{1}{10}\) can be expressed as \(10^{-1}\). Multiplying fractions like \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}\) means you're essentially multiplying \(10^{-1} \times 10^{-1} \times 10^{-1}\). Using exponent rules, you add the exponents: \(-1 + -1 + -1 = -3\), which simplifies the expression to \(10^{-3}\). This method not only simplifies the multiplication process but also helps in better understanding the nature of fractional powers and their relationships to their whole number counterparts.