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Using Properties of Exponents to Create Equivalent Expressions

Instructor: Maria Blojay

Maria has taught College Algebra and has a master's degree in Education Administration.

This lesson will show you how to use properties of exponents to create equivalent expressions. We will discuss several different properties of expressions and look at examples for how to apply each property to write expressions that are equivalent to each other.

Understanding Exponents and Equivalent Expressions

Some numbers are easy to write, such as the number 87. But others are more difficult to write down. For instance, when we multiply 87 by itself four times -- (87)(87)(87)(87) -- we get 57,289,761. Writing this long number down, or even the original equation, takes quite a bit of time, but exponents make things easier. With exponents, we can write: 874. This is called a power, where 87 is the base and 4 is the exponent.

So, we see that (87)(87)(87)(87) = 874. Keep in mind that the exponent tells us to multiply the base, 87, by itself 4 times. (It does NOT mean we multiply 87 by 4.)

Now, let's talk about different exponent properties used to create equivalent expressions, which are expressions that are equal to each other.

Properties of Exponents

Negative Exponents Property

The negative exponents property holds that a negative exponent in the numerator should be moved to the denominator and should become positive. In other words, it means that the base and exponent need to be on the other side of the fraction line and the exponent needs to be positive. So, x-a = 1/xa.

Suppose we are given this expression to evaluate:

  • 2-4.

To create an equivalent expression, you simply need to move the base and exponent to the other side of the fraction line and change the exponent to positive:

  • 1/(24)

Products of Powers

The product of powers property says that when we multiply powers with the same base, we just have to add the exponents. So, xaxb = x(a+b). As you can see, we keep the base the same and add the exponents together.

Let's try an example:

  • (x3)(x2)

Since the base is the same for each factor, we just add the exponents:

  • x(3+2) = x5

x5 is our equivalent expression. (If we wrote this out the long way, it would be: (x)(x)(x) multiplied by (x)(x).)

This property also works with negative exponents. For example, let's say that we have:

  • (x3)(x-2)

We would keep the base the same and add the exponents:

  • x(3+-2) = x1, or simply x

Quotients of Powers

The quotients of powers property says that when we divide powers with the same base, we simply subtract the exponents. So (xa/x)b = x(a-b). Notice that we keep the base the same and subtract the exponents.

Say that we have:

  • (x3) / (x2)

An equivalent expression would be:

  • x(3-2) = x1

If we wrote this out the long way, it would be (x)(x)(x) divided by (x)(x), which is simply x.

Let's look at another example:

  • (x3) / (x4)

The exponent being subtracted from is lower than the exponent being subtracted. We can still apply the quotient of powers property:

  • x(3-4), which equals x-1

Now, apply the negative exponents property to evaluate further:

  • 1/x1, which equals 1/x

Power of a Power

The power of a power property states that when we take the power of a power, we keep the base and multiply the exponents. So, (xa)b = x(a times b). This shows that we write down our base and just multiply the given exponents.

Suppose, we have this problem:

  • (x4)3

Well using the foundation presented earlier, this is the same as:

  • (x4)(x4)(x4)

Remember, when we multiply powers, we add the exponents, so:

  • x(4+4+4) = x12

Or, we could multiply the exponents of the given base. This would give us:

  • (x4)3 = x(4 times 3) = x12

Multiplying and Dividing Powers

Is it possible to multiply and divide powers in the same problem? Yes! We first simplify the powers in parentheses, then either multiply or divide the remaining problem. This is best illustrated with an example:

Say that we have:

  • x4/((x2)(x5))

To solve, first, keep the x4 in the numerator for now, then multiply the powers in the denominator by adding the exponents (since the bases are the same). So:

  • x4/ x(2+5) = x4/x7

From the remaining problem, divide the powers by subtracting the exponents:

  • x(4-7) = x-3 .

We could leave our answer this way or can rewrite the answer without negative exponent(s) by applying the negative exponents rule:

  • 1/x3

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