Question

Evaluate the power. $$ 10^{5} $$

Short answer

The value of \(10^{5}\) is 100000.

Step-by-step solution

Identify base and exponent

In the expression \(10^{5}\), the base number is 10 and the exponent is 5.

Understand exponentiation

Exponentiation means we multiply the base number by itself the number of times indicated by the exponent. In this case, we'll multiply 10 by itself 4 more times: \(10 \times 10 \times 10 \times 10 \times 10\).

Calculate the power

Perform the multiplication: \(10 \times 10 = 100\), \(100 \times 10 = 1000\), \(1000 \times 10 = 10000\) and finally, \(10000 \times 10 = 100000\). Therefore, \(10^{5}\) equals 100000.

Key concepts

Base and Exponent

When we work with exponentiation, we encounter two main components: the base and the exponent. The base is the number that is being multiplied by itself, and the exponent tells us how many times to multiply the base. For example, in the expression 10^{5}, the number 10 is the base, and 5 is the exponent. This expression tells us that we should multiply 10 by itself a total of five times.

Understanding the role of each part is crucial to evaluating powers. The exponent is not merely a number to multiply by but is rather an instruction for how many times the base is used as a factor in the multiplication. When the exponent is a positive integer, the base is multiplied by itself as many times as the value of the exponent less one because the base is already counted as the first factor.

Calculate Power

Calculating the power of a number may seem daunting at first, but it is essentially repeated multiplication. To calculate a power like 10^{5}, you start by multiplying the base (10) by itself to get 100, then multiply that result by the base to get 1,000, and continue this process the number of times indicated by the exponent (five times, in this case). This process is exponentiation, and the result is called the 'power'.

Simple Tip:

A handy tip for dealing with powers of 10 is to remember that the exponent also indicates the number of zeros in the final answer when the base is 10. Thus, for 10^{5}, the answer is 100,000 - a 1 followed by five zeros, as reflected by the exponent.

Multiplication of Numbers

Multiplication is one of the basic operations in mathematics where the concept of repeated addition is transformed into a more efficient process. When we multiply two numbers, we are essentially adding one number to itself repeatedly based on the value of the other number. This is particularly straightforward with powers of 10 because each multiplication step adds a zero to the end of the number.

Understanding and performing multiplication is fundamental to calculating powers since exponentiation is an extended form of multiplication. It's crucial to be comfortable with multiplying numbers in order to easily evaluate higher powers and work with exponentiation of various numbers.

Practice with multiplying numbers both large and small will help build a solid foundation for tackling problems involving exponentiation and ensure a stronger grasp of the overall concept.