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# Exponent Properties Review

Properties Of Exponents: We’ll look at the five prominent exemplar goods and an illustration of each. You can think of them as the order of enterprises for exponents. Learn how to handle math problems with exponents here!

We will show 8 goods of exponents. Let x and y be collections that are not equal to zero and let n and m be any integers. We also assume that no denominators are equal to zero. First, we go over each good and give examples to show how to use each property. Then, at the end of this lesson, we summarize the goods.

## Properties Of Exponents Worksheet

We are going to talk about five exponent properties. You can think about them as the order of operations for interpreters. Just like the order of transactions, you need to memorize these operations to be successful. The five exponent properties are:

• Let’s look at Product of Powers
• Power to a Power
• Quotient of potential
• Power of a Product
• Power of a result

Here’s the formula: (x^a)(x^b) = x^(a + b). When you multiply exponentials with the same base (notice that x and x are the same base), add their exponents (or powers).

Let me show you how that works. Let’s say I have (x^2)(x^3). Well, x^2 is x times x, and x^3 is x times x times x. When we add all those xs up, we get x^5, which is the same thing as adding 3 + 2.

If I have (x^2)^4, which would be x^2 proliferated four times, or x^2 times x^2 times x^2 times x^2. Once again, we add all the exponents and get x^8, and x^8 is the same as x^(2 * 4), which is 8. Not too bad, right?We can see from the formula we have (x^a)^b. When you have a power to a power, you multiply the exponents (or powers). Let me show you how this one works.

## Properties Of Exponents Calculator

In earlier chapters we introduced powers.

x3=xxx
x3=x⋅x⋅x

There are a couple of operations you can do on powers and we will introduce them now.

We can multiply powers with the same base

x4x2=(xxxx)
(xx)=x6x4⋅
x2=(x⋅x⋅x⋅x)⋅
(x⋅x)=x6

This is an example of the product of powers idiosyncrasy tells us that when you multiply powers with the same base you just have to add the exponents.

xaxb=xa+bxa⋅
xb=xa+b

We can raise a power to a power

(x2)4=(xx)
(xx)(xx)(xx)=x8(x2)4=(x⋅x)⋅
(x⋅x)⋅(x⋅x)⋅(x⋅x)=x8

This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.

When you raise a product to a power you raise each factor with a power

(xy)2=(xy)(xy)=(xx)
(yy)=x2y2(xy)2=(xy)
(xy)=(x⋅x)⋅(y⋅y)=x2y2

This is called the power of a commodity property

(xy)a=xa
ya(xy)
a=xaya

As well as we could proliferate powers we can divide powers.

x=x2

This is an example of the quotient of powers good and tells us that when you divide powers with the same base you just have to subtract the exponents.

xaxb=xab,
x0xaxb=xa−b,
x≠0

When you raise a quotient to a weight you raise both the numerator and the denominator to the power.

(xy)2=xyxy=xxy
y=x2y2(xy)2=xy⋅xy=x⋅
xy⋅y=x2y2

This is called the power of a remainder power

(xy)a=xaya,
y0(xy)a=xaya,
y≠0

When you raise a number to a zero power you’ll always get 1.

1=xaxa=xaa=x
01=xaxa=xa−a=x0
x0=1,x0x0=1,x≠0

Negative exponents are the reciprocals of the positive exponents.

xa=1xa,x0x−a=1xa,
x≠0
xa=1xa,x0xa=1x−a,
x≠0

The same properties of interpreters apply for both positive and negative exponents.

In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power

x−−√=x−−√2
=x12

## Properties Of Rational Exponents

The base a raised to the power of n is equal to the multiplication of a, n times:

a n = × a × … × a

n times

a is the base and n is the exponent.

#### Examples

31 = 3

32 = 3 × 3 = 9

33 = 3 × 3 × 3 = 27

34 = 3 × 3 × 3 × 3 = 81

35 = 3 × 3 × 3 × 3 × 3 = 243

## Division Properties Of Exponents

Remember, ‘quotient’ means ‘division’.’ The formula says (x^a) / (x^b) = x^(a – b). Basically, when you divide exponentials with the same base, you subtract the exponent (or powers).

Let me show you how this one works. Let’s say I had (x^4) / (x^3). In the top (or numerator), we have x times x times x times x. In the bottom (or denominator), we have x times x times x. Hopefully, you commemorate that x divided by x is 1, so the xs cancel. So, x divided by x is 1, x divided by x is 1, and x divided by x is 1. So, when we cancel them, what are we left with? That’s right: x^1, or just x. So (x^4) / (x^3) is just x^(4 – 3), which is x^1.

Product rule with same base

an ⋅ am = an+m

Example:

23 ⋅ 24 = 23+4 = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

Product rule with same exponent

an ⋅ bn = ( b)n

Example:

32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144

### Exponents quotient rules

Quotient rule with same base

an / am = anm

Example:

25 / 23 = 25-3 = 22 = 2⋅2 = 4

Quotient rule with same exponent

an / bn = (/ b)n

## What are the properties of exponents?

Product of a Power: When you multiply exponentials with the same infrastructure, you add their exponents (or powers). Power to a Power: When you have a power to a power, you multiply the exponents (or powers). Quotient of Powers: When you divide exponentials with the same base, you subtract the exponents (or powers).

## What are the 5 laws of exponents?

The laws of exponents are explained here along with their…
• Multiplying powers with same base.
• Dividing powers with the same base.
• Power of a power.
• Multiplying powers with the same exponents.
• Negative Exponents.
• Power with exponent zero.
• Fractional Exponent.

## What are the law of exponents?

Definition of law of exponents. : one of a set of rules in algebra: exponents of numbers are added when the numbers are multiplied, subtracted when the numbers are divided, and multiplied when raised by still another exponent: am×aⁿ=am+n; am÷aⁿ=amn; (am)ⁿ=amn.