How is combing radicals by addition and subtraction similar to combining monomial expressions by addition and subtraction? Please use examples.

Combining radicals by addition and subtraction is similar to combining monomial expressions by addition and subtraction because both involve simplifying expressions by combining like terms.

To understand this concept, let's first look at how we combine monomial expressions by addition and subtraction. Consider the following example:

3x + 2x - 5x

In this expression, we have three monomial terms: 3x, 2x, and -5x. To simplify the expression, we add or subtract the coefficients of the like terms while keeping the variable part unchanged. In this case, the like terms are 3x, 2x, and -5x because they all have the variable x.

Adding the coefficients of the like terms, we get:

(3 + 2 - 5)x

Simplifying further, we have:

0x

Since any number multiplied by zero results in zero, we can simplify the expression to:

0

Now, let's see how this process is similar when combining radicals by addition and subtraction. Consider the expression:

√2 + 3√2 - 4√2

In this expression, we have three radicals: √2, 3√2, and -4√2. To combine them, we need to look for like terms, which means the radicals must have the same radicand (the number inside the radical symbol). In this case, all the radicals have the square root of 2 as the radicand.

Adding the coefficients of the like terms, we get:

(1 + 3 - 4)√2

Simplifying further, we have:

0√2

Since 0 multiplied by any number results in zero, we can simplify the expression to:

0

So, both examples show that combining monomial expressions and radicals involves adding or subtracting the coefficients of like terms while keeping the variable (or the radicand) unchanged. The goal is to simplify the expression by grouping similar terms together.

When combining radicals by addition and subtraction, we follow similar principles as when combining monomial expressions by addition and subtraction. The main similarity lies in the fact that we need to have similar terms in order to combine them.

Let's look at an example of combining radicals by addition:
√5 + √8

In this case, we have two radicals with different radicands (5 and 8). To combine them, we need to simplify the radicals first. We can simplify √8 by breaking it down into its prime factors: √8 = √(2*2*2) = 2√2.

Now we have √5 + 2√2. This is similar to combining monomial expressions. We can think of √5 as a coefficient of √2. So, the expression can be written as √5*1 + 2√2*1.

Now, by using the distributive property to combine like terms, we get (√5*1 + 2√2*1) = √5 + 2√2.

Similarly, when subtracting radicals, we follow the same procedure of simplifying the radicals and then combining like terms. For example:
√7 - √3

We can't simplify the radicals further, so we write it as √7 - √3.

These steps are similar to combining monomial expressions. We can think of √7 and √3 as coefficients of 1. So, the expression becomes √7*1 - √3*1.

Finally, by using the distributive property to combine like terms, we get (√7*1 - √3*1) = √7 - √3.

In summary, the process of combining radicals by addition and subtraction is similar to combining monomial expressions. We simplify the radicals and then combine like terms by considering radicals as coefficients of 1 in order to perform addition and subtraction.

3 apples + 2 oranges + 2 apples - 1 orange

= 5 apples + 1 orange

3x + 2y + 2x - y
= 5x + y

3√3 + 2√5 + 2√3 - √5
= 5√3 + √5