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Introduction to Exponents

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Sophia Tutorial

This lesson introduces exponents as an operation, and discusses common and special exponents.

Tutorial

- Introduction to Exponents
- Exponents of 2 and 3
- Exponents 1 and 0

An exponent is repeated multiplication. This means that if we have the same number used in multiplication several times, we can express this using an exponent. Let's take a look at an example:

Let's examine the relationship between the 4 and the 3. We refer to 4 as the base of the expression. This is the number that is being multiplied. 3 is called the exponent, and it tells us how many times to use the base in a chain of multiplication. So we interpret as repeated multiplication: using 4 as the base in a chain of multiplication 3 times.

How do we read expressions with exponents? is read as "4 to the 3rd power." Here are some other ways to say the same thing:

So common language includes words such as "power" and "raised" and we read the base number as a normal standard number, but the exponent could be read as an ordinal number (first, second, third, etc.)

Next we are going to talk about some common exponents: 2 and 3.

**Exponents of 2 and 3**

When a base is raised to the power of 2, a common way of talking about something to the power of two is using the term "squared." Think about the area of a square. A square has side lengths that are equal to each other, and we multiply them together to find the area. So the side length *squared* gives us the area of the square.

When a base is raised to the power of 3, we commonly say that the base is "cubed." Like squaring and area, we can think about the volume of a cube. A cube has side lengths that are equal in measure, and we multiply the dimensions to find the volume. So the side length *cubed* gives us the volume of a cube.

Any base raised to the exponent 2 is called squaring the base. Any base raised to the exponent 3 is called cubing the base.

**Exponents 1 and 0**

Now let's talk about two other special exponents: one and zero. Recall that the exponent tells us how many times to use the base in a chain of multiplication. So if the exponent is 1, the expression simply equals the base, no matter what the base is.

What if the exponent is zero. A common mistake is to think that anything raised to the power of zero is zero, but this is not correct. Let's approach a zero exponent by writing another expression with an exponent, and working our way down to an exponent of zero.

As we decrease our exponent by 1, we are "losing" a factor of 4. In other words, we divide the expression above it by 4. When we get to our exponent of zero, you see we divide 1 factor of 4 by 4, leading to 1. So anything raised to the power of 0 equals 1.

Any number, variable, or expression raised to the power of 1 remains the same. Any number, variable, or expression raised to the power of zero equals 1.