Adding and Subtracting Radical Expressions With Square Roots and Cube Roots
Summary
TLDRThis 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
🧮 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.
✖️ 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.
🚀 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.
Mindmap
Keywords
💡Rational Expressions
💡Radicals
💡Like Terms
💡Simplifying Radicals
💡Distributive Property
💡Perfect Squares
💡Cube Roots
💡FOIL Method
💡Conjugates
💡Coefficients
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
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