Question

In the expression \(2^{4},\) the number 2 is called the ______________ and 4 is called the _____________

Short answer

Base, Exponent.

Step-by-step solution

Identify the Base

In the expression \(2^{4},\)\(2\) represents the number which is being multiplied. This is called the base.

Identify the Exponent

The number \(4\) in the expression \(2^{4}\) indicates how many times the base \(2\) is multiplied by itself. This is called the exponent.

Complete the Exercise

Based on the identification, the completed sentence reads: 'In the expression \(2^{4}\), the number 2 is called the base and 4 is called the exponent.'

Key concepts

Understanding the Base

The base in an expression involving exponents is the number that is being multiplied repeatedly. For example, in the expression \(2^{4}\), the base is the number 2. The base is a fundamental component because it represents the number that carries out the action of multiplication.
Without the base, there would be nothing to multiply. The base remains constant in the expression and does not change, no matter how many times multiplication occurs. The purpose of the base is to set the foundation for the multiplication process.
When you encounter an expression like \(5^{3}\), the base is 5. This means that 5 will be multiplied by itself a certain number of times, as indicated by the exponent.

Understanding the Exponent

The exponent in an expression tells you how many times to multiply the base by itself. For example, in \(2^{4}\), the exponent is 4. This means you multiply the base (2) four times: \(2 \times 2 \times 2 \times 2\).
The exponent is a critical part of the expression because it indicates the number of repetitive multiplications. It controls the magnitude of the output. The larger the exponent, the more times the base number is multiplied by itself.
It's important to note that an exponent of 1 means the base is taken just once, e.g., \(3^{1} = 3\), while an exponent of 0 means the result is always 1, e.g., \(3^{0} = 1\). This rule applies to any base except zero.

The Role of Multiplication in Exponents

Multiplication is the operation carried out repeatedly when working with exponents. In the expression \(2^{4}\), the base 2 is multiplied by itself four times, resulting in \(2 \times 2 \times 2 \times 2 = 16\).
One way to think about this is to consider exponents as a shortcut for writing long multiplication sequences. Instead of writing \(2 \times 2 \times 2 \times 2\), you can simply write \(2^{4}\).
Additionally, understanding the relationship between multiplication and exponents can help simplify complex problems. For example, \((2^{3})^{2}\) can be transformed using multiplication into \(2^{3 \times 2} = 2^{6}\), showing how different properties of exponents leverage multiplication. This makes handling large and complex numbers more manageable.