Question

Multiply. $$\left(10^{5}\right)\left(10^{9}\right)$$

Short answer

\(10^{5} \times 10^{9} = 10^{14}\)

Step-by-step solution

Understand the Law of Exponents for Multiplication

When you multiply two exponents with the same base, you add the exponents. Here, the base is 10, and you have two exponents: 5 and 9.

Add the Exponents

Add the exponent values: 5 (from the first term) + 9 (from the second term) = 14.

Apply the Sum of Exponents

Write the base 10 raised to the power of the sum of the exponents: 10 raised to the 14th power, or \(10^{14}\).

Key concepts

Exponential Notation

Exponential notation is a concise way to represent repeated multiplication of the same number. It consists of a base and an exponent. The base is the number being multiplied, and the exponent, written as a superscript, tells you how many times the base is used as a factor in the multiplication. For example, in the expression (10^{5}), 10 is the base and 5 is the exponent, indicating that 10 is multiplied by itself 5 times.

When numbers are expressed exponentially, it makes it much easier to handle large numbers or very small numbers, which is especially useful in fields like science and engineering. Instead of writing out lengthy multiplications, such as 10 * 10 * 10 * 10 * 10, we can simply write (10^{5}). This form is not only more compact but also simplifies many mathematical operations with rules such as those for multiplying exponents.

Multiplying Exponents

When dealing with the multiplication of exponents that have the same base, the Law of Exponents provides a very efficient method to simplify the expression. According to this law, rather than multiplying the bases multiple times, we simply add the exponents together. This technique of multiplying exponents is a fundamental concept that makes dealing with exponential expressions much more manageable.

To illustrate, consider the multiplication of (10^{5}) and (10^{9}). Both have the same base of 10. By applying the Law of Exponents for multiplication, we add the exponents 5 and 9 to get 14, resulting in (10^{14}). It's vital to remember that this rule only applies when the bases are identical. Students learning this concept should practice with various bases and exponents to become comfortable with the operation.

Mathematical Operations

Mathematical operations are the foundation of mathematics and include addition, subtraction, multiplication, and division. Beyond these basic operations, there are more advanced ones like exponentiation, which involves exponents, and rules governing these processes to simplify complex expressions. A solid understanding of these rules, such as the Law of Exponents, aids in carrying out operations correctly and efficiently.

For a student to truly grasp and apply the Law of Exponents, it's important to practice with a variety of expressions. This practice helps to build a strong foundation in mathematical operations, preparing students for more complicated scenarios such as working with variables, higher math, and practical applications in science and finance. Remember that each operation has its own set of rules; knowing them can significantly improve your mathematical skills and confidence in handling various problems.