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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.

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Introduction to Exponents

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.