Question

Evaluate expression. $$2^{5}$$

Short answer

The value of \(2^{5}\) is 32.

Step-by-step solution

Understand the Expression

The expression given is an exponentiation, which means we need to multiply the base, which is 2, by itself as many times as the exponent, which in this case is 5.

Calculate the Value

We multiply 2 by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2\).

Key concepts

Powers of Numbers

Powers of numbers are an elegant way to express repeated multiplication of the same factor. When we say a number is 'raised to a certain power,' we mean it's multiplied by itself a set number of times. The number that is being multiplied is called the 'base,' and the number of times it is multiplied is known as the 'exponent' or 'power'.

For instance, take the expression \(2^5\), which involves the base 2 and the exponent 5. It simply means 2 multiplied by itself 4 more times. This can be visualized as \(2 \times 2 \times 2 \times 2 \times 2\). The exponent here tells us we have a total of five 2's, and we need to perform the multiplication step by step.

Exponential Expressions

Exponential expressions, such as \(2^5\), compactly capture the process of scaling numbers. They form the building blocks for more advanced topics, like exponential growth or decay in the realms of science and finance. The ability to understand and manipulate exponential expressions is crucial when dealing with large quantities or rates of increase/decrease.

An exponential expression consists of two main components: the base and the exponent. The base is the number being multiplied, and the exponent indicates how many times to use the base as a factor. In improving our understanding of these expressions, it is essential to remember that any number, except zero, raised to the power of zero is always 1, and any number raised to the power of 1 is the number itself.

Basic Arithmetic Operations

Basic arithmetic operations, such as addition, subtraction, multiplication, and division, are the foundational tools we use in mathematics to solve a variety of problems. Exponentiation, which we see in the expression \(2^5\), is a form of repeated multiplication, one of these basic operations.

In solving for the value of \(2^5\), we perform the multiplication operation multiple times. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us through complex calculations. Here, we process exponents right after any tasks within parentheses. Understanding and correctly applying these operations allow us to navigate through more intricate mathematical expressions effectively.