Question

In the expression \(5^{4}\) state which number is the index and which is the base.

Short answer

Answer: In the expression \(5^4\), the base is 5 and the index (exponent) is 4.

Step-by-step solution

Identify the base

In the expression \(5^4\), the base is the number 5, which is the one being raised to a power.

Identify the index (exponent)

In the expression \(5^4\), the index (or exponent) is the number 4, which indicates the number of times the base (5) is multiplied by itself. So, in the expression \(5^4\), the base is 5 and the index (exponent) is 4.

Key concepts

Base in Exponentiation

When we come across terms like exponentiation, the term base plays an important role. Think of the base as the foundation of a building. In exponentiation, the base is the number that is being multiplied by itself. It's the main ingredient in the recipe for powers. In the expression \(5^{4}\), for instance, the base is 5. This means that we start with the number 5, and the process of exponentiation will involve this number in a critical way.

Understanding what the base is, is crucial because it defines what will be repeated in the operation. If you change the base, you change the whole outcome of the exponentiation. So, always make sure you correctly identify the base first, because everything in the exponentiation process relies on it.

Remember that in any expression like \(a^{n}\), where a is any given number and n is the natural number representing the exponent, 'a' is the base. Even if 'a' is negative, a fraction, or even another power, it still remains the heart of the operation, the number that we'll be using as the starting point for our multiplication journey.

Index or Exponent

After identifying the base, we encounter the index, also known as the exponent, which is an equally critical component in exponentiation. This is the small number that hovers to the upper right of the base. The job of the exponent is to tell us how many times the base should be multiplied by itself. It is essentially an instruction - a repetitive directive.

For example, in \(5^{4}\), while we've established that 5 is the base, the number 4 is the exponent. This indicates that you should multiply 5 by itself a total of four times: \(5 \times 5 \times 5 \times 5\). The reason we have exponents is that they provide a shortcut for writing that multiplication out; instead of a long string of the base repeated multiple times, we have a clean, compact notation. It's a classic case of mathematics making life easier!

Reading Exponents

Furthermore, understanding exponents is not just about recognizing them in an expression, but also about knowing how to read and interpret them. The expression \(5^{4}\) is read as 'five to the fourth power' or simply 'five to the fourth'. This linguistic simplification goes hand in hand with the notational simplicity exponents provide.

Multiplication of a Number by Itself

The core idea behind exponents is the multiplication of a number by itself. This repetitious multiplication is something you've been doing since you learned your times tables, but with exponents, it becomes a more abstract and powerful tool.

Take \(5^{4}\) from our original exercise. What we're doing here is taking the number 5 and multiplying it by itself exactly 4 times. This is not the same as multiplying 5 by 4; this is a successive multiplication where each product is used as one of the multiplicands for the next iteration: \( 5 \times 5 = 25 \), then \(25 \times 5 = 125 \), and lastly \(125 \times 5 = 625\). It is a testament to the power of exponentiation that we can succinctly express such a potentially vast multiplication process with just two numbers: the base and the exponent.

Multiplication of a number by itself is a cornerstone concept in mathematics as it forms the foundation of powers and roots, and it's all about understanding and visualizing the growth pattern of numbers. With consistent practice and visualization, grasping this multiplication system becomes intuitive, and soon, you will be able to 'see' the pattern of multiplication quickly in your mind's eye, enabling a deeper comprehension and quicker calculations.