Question

Multiply the following powers of \(10 .\) $$10^{-2} \cdot 10^{5}$$

Short answer

\(10^{-2} \times 10^{5} = 10^{3}\)

Step-by-step solution

Identify the Laws of Exponents for Multiplication

When multiplying two powers that have the same base, you can add the exponents according to the law: \(a^m \times a^n = a^{m+n}\). Here, our base is 10.

Add the Exponents

Since both powers have the base 10, add the exponents: \(-2 + 5 = 3\).

Write the Final Answer

After adding the exponents, the answer is \(10^{3}\).

Key concepts

Multiplying Powers of 10

Multiplying powers of 10 is a fundamental concept in mathematics, especially when dealing with large or small numbers. The process is straightforward: when you are multiplying powers of 10, you simply need to add the exponents. This is based on the law of exponents for multiplication, which states that for any base 'a' (in this case 10), the multiplication of two powers with the same base can be simplified by adding their exponents.

For example, consider the expression \(10^{-2} \cdot 10^{5}\). To multiply these two powers of 10, you add the exponents: \( -2 + 5 = 3 \). Thus, the result of the multiplication is \(10^{3}\), which equals 1,000. This process beautifully illustrates the simplicity of working with powers of 10; each multiplication step is a mere addition of exponents.

Exponent Addition

Exponent addition is one of the basic laws of exponents that applies when you are multiplying like bases. This rule is crucial for simplifying expressions and solving equations involving exponentiation. To understand this concept, remember that the exponent denotes how many times the base is used as a factor.

Here's how it works: when you multiply \(a^m\) by \(a^n\), you're actually multiplying 'a' by itself 'm' times and then 'n' times, making it a total of \(m + n\) times that 'a' is used as a factor. Therefore, \(a^m \times a^n = a^{m+n}\). For instance, with our earlier example of \(10^{-2} \cdot 10^{5}\), instead of multiplying ten times itself -2 times, and then 5 times, you can directly sum the exponents to get \(10^{3}\), highlighting the power of exponent addition for simplification.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise and standardized form, making them easier to work with. This notation uses the power of ten to represent numbers as a product of two parts: a coefficient, which is usually a value between 1 and 10 (but not inclusive of 10), and \(10\) raised to a power, which indicates the number of decimal places the decimal should move.

Scientific notation is perfect for simplifying calculations with very large or very small numbers because it reduces complex numbers to a base (usually 10) and a manageable exponent. For example, the mass of the Earth might be written as \(5.972 \times 10^{24} \text{ kg}\) in scientific notation, instead of writing out all 24 zeros. When performing operations with numbers in scientific notation, the laws of exponents, like multiplying powers of 10 and exponent addition, are immediately applicable and streamline the process greatly.