Adding and Subtracting Radical Expressions With Square Roots and Cube Roots

Summary

This lesson covers the addition, subtraction, and multiplication of radical expressions. It emphasizes how like terms are essential for adding or subtracting radicals and demonstrates the process of simplifying square and cube roots. The video provides several examples of combining radicals, breaking them down using perfect squares or cubes, and solving equations with distributed terms. Through various problem-solving steps, the video illustrates the importance of simplifying radicals and recognizing when terms can be combined, concluding with an advanced example involving the multiplication of conjugates and expanded expressions.

Takeaways

• 🔢 You can add or subtract radical expressions if they have the same radicals (like terms), such as combining 4√5 + 6√5 into 10√5.
• ❌ Expressions like 4√3 + 6√5 cannot be combined because the radicals are different.
• ➕ When adding or subtracting radical expressions with the same radical, only the coefficients are combined, like simplifying 7√2 - 3√2 + 5√2 into 9√2.
• 🟢 Simplifying radicals can make expressions combinable, such as breaking down √8 and √18 to form like terms √2, allowing the expressions to be combined.
• ✂️ Radicals like √12, √27, and √48 can be simplified into common radicals (like √3), which allows terms to be combined, yielding a result like 9√3.
• 🧮 Cube roots can be handled similarly, and simplification to like terms (such as cube root of 2) makes combining expressions possible.
• 🔄 Distributive property applies when multiplying radical expressions, such as in (√3)(7 + √3), which simplifies through distribution.
• 🧩 Multiplying conjugates (like 4 - √6 and 4 + √6) eliminates the middle terms, simplifying the result.
• 📐 For expressions like (2 + √3)², fully expanding by using FOIL helps combine the middle terms and results in simplified answers.
• 🔢 Complex problems involving radicals raised to powers can be simplified step-by-step using FOIL and multiplication, as shown with (4√3 + 2)³.

Q & A

What is the sum of 4√5 and 6√5?

Since both terms have the same radical (√5), you can add the coefficients, 4 and 6. The result is 10√5.

Why can't you add 4√3 and 6√5?

You can't add 4√3 and 6√5 because their radicals (√3 and √5) are different. You can only combine terms with the same radical.

How would you simplify the expression 7√2 - 3√2 + 5√2?

Since all the terms have the same radical (√2), you can combine the coefficients. 7 - 3 + 5 equals 9, so the answer is 9√2.

How do you simplify 3√8 - 5√18?

First, simplify the radicals: √8 becomes 2√2, and √18 becomes 3√2. After that, 3(2√2) = 6√2 and 5(3√2) = 15√2. Finally, subtract: 6√2 - 15√2 = -9√2.

What is the simplified form of 4√12 + 3√27 - 2√48?

Simplify the radicals: √12 becomes 2√3, √27 becomes 3√3, and √48 becomes 4√3. Then, the expression becomes 8√3 + 9√3 - 8√3. The result is 9√3.

How do you simplify cube roots in expressions like 16^(1/3), 54^(1/3), and 128^(1/3)?

You break the numbers down into perfect cubes. For example, 16 = 8 * 2, 54 = 27 * 2, and 128 = 64 * 2. Simplify the cube roots, and since all terms share a common radical (³√2), combine the coefficients.

What is the result of multiplying √3 by (7 + √3)?

Distribute √3: √3 * 7 = 7√3, and √3 * √3 = 3. So, the final answer is 7√3 + 3.

How do you simplify the expression 4√5 * √7 - √3?

First, multiply 4√5 by √7, which gives 4√35. Then multiply 4√5 by √3, which gives √15. The final expression is 4√35 - √15.

What is the result of multiplying conjugates like (4 - √6)(4 + √6)?

The middle terms cancel, leaving only 4² - (√6)². This simplifies to 16 - 6, which equals 10.

How do you expand and simplify (5 + √2)²?

First, apply the distributive property: (5 + √2)(5 + √2). This results in 25 + 10√2 + 2, which simplifies to 27 + 10√2.

Outlines

00:00

🧮 Simplifying Radicals: Adding and Subtracting Like Terms

This paragraph explains how to add and subtract radicals. If the radicals are the same, such as 4√5 + 6√5, you can add the coefficients, yielding 10√5. However, if the radicals differ, like 4√3 + 6√5, they cannot be combined. The section continues with examples, highlighting how to simplify and combine radicals when possible, such as 7√2 - 3√2 + 5√2. Through step-by-step explanations, the paragraph shows that the final result is 9√2, and introduces more complex problems involving radicals and their simplification.

05:03

✖️ Distributing Radicals and Multiplying Conjugates

This section focuses on distributing radicals and multiplying expressions involving radicals. It starts with an example of distributing √3 across a sum, then proceeds to more complex operations like multiplying conjugates (e.g., (4 - √6)(4 + √6)) and the steps to simplify them. The middle terms in such products cancel out, leaving simplified results like 10. The paragraph emphasizes key algebraic techniques such as FOIL and distribution while applying them to both basic and advanced radical problems.

10:03

🚀 Expanding and Simplifying Radical Expressions

This paragraph covers the expansion and simplification of more complicated expressions involving radicals, particularly when terms are squared or cubed. Examples like (2 + √3)² and 5 + √2² are worked through, showing the full expansion and combination of like terms. The final example involves expanding (4√3 + 2)³, where the process includes multiple steps of FOIL and distribution, leading to a final answer of 296 + 240√3. The paragraph provides a detailed walkthrough of how to handle nested radical expressions.

Keywords

💡Rational Expressions

Rational expressions are algebraic expressions involving fractions with polynomials in the numerator and denominator. In the video, the speaker discusses how to add and subtract rational expressions, emphasizing the need for common radicals when dealing with such operations.

💡Radicals

A radical refers to an expression that includes a square root, cube root, or other root. The video focuses on the manipulation of square roots and cube roots, explaining how to simplify and combine terms that include radicals by ensuring they have like radicals before performing operations.

💡Like Terms

Like terms are terms in an expression that have the same variables or radical parts, which allows them to be combined. In the context of the video, the speaker highlights that terms with the same radical, such as 'root 5', can be added or subtracted by operating on their coefficients.

💡Simplifying Radicals

Simplifying radicals involves breaking down a number under a radical into factors, where at least one factor is a perfect square (or cube, etc.), so it can be simplified. The video demonstrates this process, for example, breaking down 'root 48' into '4 * root 3' and simplifying it.

💡Distributive Property

The distributive property allows for multiplying a single term across terms within parentheses. In the video, the property is used in problems like 'root 3 * (7 + root 3)', where the speaker explains how to distribute the radical across both terms inside the parentheses.

💡Perfect Squares

A perfect square is a number that can be expressed as the square of an integer, such as 4 or 9. In the video, perfect squares are used to simplify radicals by extracting the square root of the number, such as turning 'root 4' into '2' and 'root 9' into '3'.

💡Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. The speaker introduces cube roots in the video when simplifying expressions like 'cube root of 8', and relates how to combine cube roots when they have the same radical part.

💡FOIL Method

The FOIL method refers to a specific technique for multiplying two binomials, using the First, Outer, Inner, and Last terms. The speaker uses this method when multiplying conjugates and other expressions like '(5 + root 2) * (5 - root 2)' to simplify them step by step.

💡Conjugates

Conjugates are pairs of binomials that differ only in the sign between two terms, like '(a + b)' and '(a - b)'. In the video, conjugates are multiplied using the FOIL method, with the result that the middle terms cancel out, simplifying the expression.

💡Coefficients

A coefficient is the numerical factor that multiplies a variable or a radical in an expression. In the video, the speaker frequently refers to adding or subtracting coefficients when the radicals are the same, such as adding '4 root 5' and '6 root 5' by combining the coefficients 4 and 6 to get '10 root 5'.

Highlights

Adding and subtracting radical expressions requires like terms (same radicals).

4√5 + 6√5 can be simplified by adding the coefficients, giving 10√5.

You cannot combine radicals with different roots, e.g., 4√3 + 6√5 remains unchanged.

7√2 - 3√2 + 5√2 simplifies to 9√2 by adding the coefficients.

Simplifying square roots can sometimes allow for combining like terms, e.g., 3√8 - 5√18 becomes -9√2.

To simplify roots like 4√12 + 3√27 - 2√48, break down the radicals into perfect squares and simplify.

In the problem 4√12 + 3√27 - 2√48, the final answer is 9√3 after simplifying.

When working with cube roots, break down the numbers into perfect cubes and simplify.

Multiplying square roots follows distribution, e.g., √3(7 + √3) simplifies to 7√3 + 3.

When multiplying terms like 4√5 * √7, combine the numbers inside the square roots (5 * 7 = 35).

Multiplying conjugates (e.g., (3 - √5)(3 + √5)) results in the middle terms canceling out, leaving a simplified difference.

Squaring binomials like (2 + √3)² requires using FOIL method for full expansion.

In the problem (2 + √3)², the expanded form is 7 + 4√3 after adding like terms.

To expand and simplify higher powers (e.g., (4√3 + 2)³), break it down into steps and use FOIL.

Final result of (4√3 + 2)³ is 296 + 240√3 after expanding and combining terms.

Transcripts

in this lesson we're going to talk about

adding and subtracting rational

expressions

what is 4

root 5 plus 6 root 5

what is that equal to

now because the radicals are the same

you can add the coefficient in front of

it that is you can add four

plus six

which is ten

so this is equal to ten root five

let's say if you have four radical three

plus six radical five

you cannot add four plus six

because what would you do would you say

it's ten root three

or ten root five it doesn't work because

these two are not the same

you can only add them if they're the

same

for example you can add 4x plus 6x

that's 10x because they're like terms

but you can't say 4x plus 6y

is 10xy

it just doesn't work see you only can

add or subtract like terms

so knowing that

go ahead and simplify this problem

7 root 2

minus 3 root 2

plus 5 root two

so all of these are like terms they all

contain the square root of two

seven minus three is four

and four plus five is nine

so it's nine root two

now what about this one what's 3

times the square root of 8

minus 5

times

the square root of 18.

right now we cannot combine them because

8 and 18 are not the same

but sometimes

we can simplify the radicals and make it

the same

eight is basically four times two

eighteen is nine times two

the square root of four is two

and the square root of nine is three

three times two is six

five times three is fifteen

now notice that we have like terms

so now we can combine them

six minus fifteen is negative nine

so the final answer is negative nine

root two

try this one

four root twelve

plus 3 root 27

minus 2 root 48

feel free to pause the video and work on

this example

so 12 we can break that into 4 and 3

27 a perfect square that goes into 27 is

9

so 9 and 3

and 48

contains a perfect square which is 16.

notice that we have a common radical

root three

so on the right track

the square root of four is two

the square root of nine

is three

and the square root of sixteen is four

four times two is eight

three times three is nine

two times four is eight

so eight and negative eight they cancel

they add up to zero

so the final answer is just going to be

9 root 3.

let's try one more example but this time

we're going to use cube roots

go ahead and try that

so think of the perfect cubes 8 27 64

125

8 goes into 16

so you want to break down 16 as 8 and 2

27 goes into 54

so 54 divided by 27 is two

and 64 goes into 128 64 times 2

is 128. notice that every term has a

cube root of 2.

so these are all like terms the cube

root of 8 can be simplified to 2

and the cube root of 27

is 3

and the cube root of 64

is 4.

so 8 times 2 is 16.

five times three is fifteen

and three times four is twelve

now

fifteen minus twelve

is three

and sixteen plus three is nineteen so

the answer is nineteen cube root two

if you were to see a homo problem that

looks like this

what would you do what is the square

root of three

times

seven plus root three

feel free to try that problem

so here you have to distribute

root three times seven

is simply seven root three

and root three times itself

three times three is nine so that's the

square root of nine

and the square root of nine is three

so that's the answer you just gotta

distribute try this one what is four

root five

times root seven

minus the square root of three

feel free to pause the video and work on

that example so let's multiply the first

two

we can't multiply 4 and 7 because the 7

is inside the radical and the 4 is on

the outside

however we can multiply 5 and 7

because they're both inside the square

root function

5 times 7 is 35

and then let's multiply 4 root 5 by root

3.

so 5 times 3 is 15 and that's going to

be inside the radical

and so that's the answer

now sometimes

you might be given a problem where you

have to multiply

two conjugates together

whenever you see this

you need to foil and know that the

middle terms will cancel

so first it's going to be 3 times 3

which is 9

and then we're going to multiply 3 times

negative root 5

that's negative 3 root 5

and then root 5 times 3

so that's positive 3 root 5

and the square root of 5 times negative

root 5 that's the negative root 25 which

is negative 5.

the two middle terms will cancel they

add up to 0

and 9 minus five is equal to four

let's try another example

four minus root six

times four plus root six

so we already know that the middle terms

will cancel so all we need to do is

multiply the outer terms

4 times 4

is 16

and negative root 6 times root 6

that's negative root 36 which is

negative 6

and sixteen minus six is ten that's

going to be the final answer

now what about this one what is two plus

root three squared

if you see it you need to expand it you

have to multiply it by itself two times

now in this example these are not

conjugates of each other

because the sign is the same

therefore the middle terms will not

cancel

so we need to foil completely

2 times two is four

two times root three that's

positive two root three

and then root three times two that's

also two root three

and root three times root three is just

three

so let's add the middle terms

two plus two is uh four so that's going

to be four root three

and four plus three is seven so it's

seven plus four

root three

let's try one more example

5 plus root 2 squared

so five times five is twenty-five

we know the two middle terms will be

five root two

and the last one

that's simply going to be two

five plus 5 is 10

and 25 plus 2 is 27

and so this is the answer

so here's another example that we can

try

go ahead and expand it and simplify

so 4 square root 3 plus 2 raised to the

third power

so we can write this

three times

and i'm running out of space

as always

but first let's foil the first two

so 4 times 4

that's 16

and the square root of 3 times the

square root of 3 that's the square root

of 9 which is 3

and then we have 4 times 2 which is 8

and the square root of 3 will be carried

over and this is also 8 square root 3

and then 2 times 2 is 4.

now we still have another one that we're

going to have to multiply later but

let's just

write this for now

so 16 times 3 is 48 48 plus 4

is 52

and 8 plus 8 that's 16 so we've got 16

square root 3.

so now

we need to foil these two expressions

so now 52 times 4

50 times 4 is 200 and 2 times 4 is 8 so

that's going to be 208 square root 3 and

then 52 times 2

that's 104

and here we have 16 times 4

which is going to be 64.

and then

square root 3 times square root 3 that's

3

and then 16 times 2 is 32 but it's going

to be 32 square root of 3.

now 64 times 3 is 192. if we add that to

104

that's 296.

then we could add these two

so 208 plus 32 that's going to be 240.

so this is the final answer

296 plus 240 square root 3.

you