Question

In the expression \(15^{3},\) what is 15 called? What is 3 called? What is the expression called?

Short answer

In the expression, \(15^{3}\), 15 is called the base, 3 is known as the exponent and the whole expression is called an exponential expression.

Step-by-step solution

Identifying the Base

The base of this exponential expression is the number which is getting multiplied. Here, the base is 15.

Identifying the Exponent

The exponent in this exponential expression is the number which dictates how many times the base is multiplied by itself. In this case, the exponent is 3.

Identifying the Expression

The overall term or notation used here is that of an exponential expression. This is due to the format in which it's written: a base (in this case, 15) being 'raised to the power of' an exponent (3 in this instance).

Key concepts

Base in Exponents

Understanding the base in exponents is crucial when dealing with exponential expressions. In mathematical terms, the base is the number that is to be multiplied by itself a certain number of times as instructed by the exponent. Take the exponential expression from the exercise, \(15^{3}\), for example. Here, the number 15 acts as the base. It is the 'starting point' from which you'll build up the final value.

Think of the base as the foundation of a building. Just as a strong foundation determines the stability of a building, knowing the base is essential for understanding the full exponential expression. It's also worth noting that any non-zero base raised to the power of zero equals 1, illustrating that the base is a fundamental component of the expression, no matter what the exponent may be.

Exponent

An exponent is a small but powerful number placed to the upper-right of the base, dictating how many times the base will be multiplied by itself. This small numeral can significantly increase the value of an expression, as it represents repeated multiplication. In our example \(15^{3}\), the exponent is 3, which means 15 must be multiplied by itself a total of three times, i.e., \(15 \times 15 \times 15\).

The exponent can be thought of as a shorthand for a long multiplication process. Without exponents, writing out repeated multiplication would be cumbersome, especially for large numbers or numerous repetitions. They make lengthy calculations more manageable and are a testament to the elegance and shorthand of mathematical notation.

Powers of Numbers

When we raise a base to an exponent, we are taking a number to its power, which is a fundamental concept in algebra and higher mathematics. The term 'power' refers to the overall product that results from the exponential notation. For instance, \(15^{3}\) is read as 'fifteen raised to the third power' or simply 'fifteen to the power of three'.

Each power of a number is notably different and reflects exponential growth. If you observe the progression of \(15^{1}\), \(15^{2}\), \(15^{3}\), and so forth, you'll see the outcome is vastly different with each increment of the exponent, showcasing the explosive nature of exponential growth. Remember, the power of a number reveals just how multiplication scales in a rapid and significant way.

Multiplication of a Number by Itself

Multiplication of a number by itself is the heart of what an exponent expresses. To say \(15^{3}\) is to say 15 is used as a factor three times in a multiplication operation: \(15 \cdot 15 \cdot 15\). This is different from simply tripling a number, where you add it to itself twice (15 + 15 + 15). Multiplication by itself exponentially increases a number's value, accentuating the difference between addition and multiplication.

Here's a helpful visualization: If you had 15 coins and created stacks of coins, each with 15 in them, three different times, you would be performing an operation akin to the concept of \(15^{3}\). It illustrates how a number grows not in a straight line, as with addition, but in a much more dramatic curve, showing the compounding effect of multiplication.