exponents and radicals class 8

. WHAT IS AN EXPONENT?

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An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 2³) means:

2 x 2 x 2 = 8.

2³ is not the same as 2 x 3 = 6.

Remember that a number raised to the power of 1 is itself. For example,

a¹ = a

5¹ = 5.

 

There are some special cases:

1. a⁰ = 1

When an exponent is zero, as in 6⁰, the expression is always equal to 1.

a⁰ = 1

6⁰ = 1

14,356⁰ = 1

2. a^-m = 1 / a^m

When an exponent is a negative number, the result is always a fraction. Fractions consist of a numerator over a denominator. In this instance, the numerator is always 1. To find the denominator, pretend that the negative exponent is positive, and raise the number to that power, like this:

a^-m = 1 / a^m

6^-3 = 1 / 6³

You can have a variable to a given power, such as a³, which would mean a x a x a. You can also have a number to a variable power, such as 2^m, which would mean 2 multiplied by itself m times. We will deal with that in a little while.

First let's look at how to work with variables to a given power, such as a³.

There are five rules for working with exponents:

1. a^m * aⁿ = a^(m+n)

2. (a * b)ⁿ = aⁿ * bⁿ

3. (a^m)ⁿ = a^{(m * n)}

4. a^m / aⁿ = a^(m-n)

5. (a/b)ⁿ = aⁿ / bⁿ

 

Let's look at each of these in detail. EXPLAINATION IN DETAILS

1. a^m * aⁿ = a^(m+n) says that when you take a number, a, multiplied by itself m times, and multiply that by the same number a multiplied by itself n times, it's the same as taking that number a and raising it to a power equal to the sum of m + n.

Here's an example where

a = 3
m = 4
n = 5

a^m * aⁿ = a^(m+n)

3⁴ * 3⁵ = 3⁽⁴⁺⁵⁾ = 3⁹ 

 

2. (a * b)ⁿ = aⁿ * bⁿ says that when you multiply two numbers, and then multiply that product by itself n times, it's the same as multiplying the first number by itself n times and multiplying that by the second number multiplied by itself n times.

Let's work out an example where

a = 3
b = 6
n = 5

(a * b)ⁿ = aⁿ * bⁿ

(3 * 6)⁵ = 3⁵ * 6⁵

18⁵ = 3⁵ * 6⁵ 

 

3. (a^m)ⁿ = a^{(m * n)} says that when you take a number, a , and multiply it by itself m times, then multiply that product by itself n times, it's the same as multiplying the number a by itself m * n times.

Let's work out an example where

a = 3
m = 4
n = 5

(a^m)ⁿ = a^{(m * n)}

(3⁴)⁵ = 3^{(4 * 5)} = 3²⁰ 

 

4. a^m / aⁿ = a^(m-n) says that when you take a number, a, and multiply it by itself m times, then divide that product by a multiplied by itself n times, it's the same as a multiplied by itself m-n times.

Here's an example where

a = 3
m = 4
n = 5

a^m / aⁿ = a^(m-n)

3⁴ / 3⁵ = 3^(4-5) = 3^-1 (Remember how to raise a number to a negative exponent.)

3⁴ / 3⁵ = 1 / 3¹ = 1/3

 

5. (a/b)ⁿ = aⁿ / bⁿ says that when you divide a number, a by another number, b, and then multiply that quotient by itself n times, it is the same as multiplying the number by itself n times and then dividing that product by the number b multiplied by itself n times.

Let's work out an example where

a = 3
b = 6
n = 5

(a/b)ⁿ = aⁿ / bⁿ

(3/6)⁵ = 3⁵ / 6⁵

Remember 3/6 can be reduced to 1/2. So we have:

#what is radicals