Radical Expressions

Question 1

an expression or quantity that has the radical symbol or uses a root (√).

Answer 1

Radical

Question 2

Example:Evaluate the following roots:

√49
∛125
√121

Answer 2

Solution:

√49 is equal to7since 7 x 7 = 49;
∛125is equal to5since 5 x 5 x 5 = 125; and
√121is equal to11since 11 x 11 = 121.

Question 3

A radical expression has three parts or components:

Answer 3

the radical symbol, the radicand, and the index or degree of the radical.

Question 4

the symbol that indicates we are taking the root of a number.

Answer 4

Radical sign

Question 5

the quantity inside the radical sign. It is the one that you are taking the root of.

Answer 5

Radicand

Question 6

This is the tiny number that you can see on the upper left side of the radical sign. This number tells us how many times we should multiply the resulting number to obtain the radicand

Answer 6

index or degree of the radical

Question 7

Example:Determine the radicand and index of the following radical expressions:

1).√x+8
2).√a⁵-2
3).⁴√10
4).³√y²
5).2√3

Answer 7

Solution:

The radicand isx + 8while the index is 2 since the index is missing in the radical symbol.The radicand isa5– 2while the index is 2.The radicand is 10 while the index is 4.The radicand isywhile the index is 3.The radicand is 3 while the index is 2. Take note that the 2 outside the radical sign is not part of the radicand.

Question 8

Example 1:Write√15 as an expression with fractional exponents

Answer 8

Solution:The index of √15 is 2 and we have 1 as the power of the radicand. Therefore, our fractional exponent is ½. Thus, √15= 151/2.

Question 9

Example 4:Write a3/4as a radical expression.

Answer 9

Solution:The denominator of the fractional exponent ofa3/4is 4. This means that the index of our radical is 4. Meanwhile, the numerator of the fractional exponent is 3. Hence, it will be the power of our radicand. Therefore, a3/4is equal to ∜a³.

Question 10

Example 5:Write (y – 1)⅓as a radical expression.

Answer 10

Solution:The denominator of the fractional exponent is 3. This means that the index of our radical is 3. Meanwhile, the numerator of the fractional exponent is 1. Hence 1 will be the power of our radicand. Therefore, (y – 1)⅓= ∛(y – 1).

Question 11

Suppose we want to write ∜b2as an expression with a fractional exponent.

Answer 11

However, take note that we can reduce 2⁄4 into its lowest terms which is ½. Thus, b2/4is also equivalent to b1/2.

Therefore, ∜b2= b^2/4= b^1/2

Question 12

If the index of the radical and the power of the radicand are equal, then the radical sign will cancel out, leaving us with only the radicand.

Answer 12

Property #1

Question 13

What is the vvalue of ⁸√(9 - x)⁸

Answer 13

=9-x

Question 14

tells us that the root of the product of given numbers is equal to the product of the roots of the given numbers.

Answer 14

Property #2

Question 15

tells us that the root of the ratio of two numbers is equal to the ratio of the roots of two numbers.

Answer 15

Property #3

Question 16

A radical expression is in its simplified/simplest form if all of the following conditions are met:

Answer 16

all exponents of its radicand has no common factor with the index of the radicalall exponents of its radicand are less than the index of the radicalthe radicand has no fractions involvedthere is no radical in the denominatorIn the case of square root or cube root of a number:the radicand has no factor that is a perfect square number or a perfect cube number (we will discuss this later).

Question 17

For example, √x is a radical expression in simplified form because:

Answer 17

The exponent of its radicand (which is 1) has no common factor with the index (which is 2).The exponent of its radicand (which is 1) is less than the index (which is 2).The radicand of √x has no fractions involved since the radicand is justx.There is no radical in the denominator of √x since the denominator of √x is simply 1.We are dealing with square root andxis not a perfect square quantity.

Question 18

Example:Determine which of the following radical expressions is/are in simplified form:

1).√x⁵
2).¹⁵√q⁵
3).1/√x
4.√a-b

Answer 18

Solution:

The radical expression in item 1 is not in simplified form since the power of the radicand (which is 5) is greater than the index of the radical (which is 2).The radical expression in item 2 is not in simplified form since the radicand and the index have a common factor. The common factor of 5 (exponent of the radicand) and 15 (index) is 5.The radical expression in item 3 is also not in simplified form since there’s a radical in the denominator.The radical expression in item 4 is the only radical in simplified form. It satisfies the four conditions of the simplified form of a radical expression.

Question 19

are numbers with a whole number cube root. For example, 8 is a perfect cube since ∛8 = 2.

Answer 19

Perfect cube numbers

Question 20

Example 1:Simplify√50

Answer 20

Solution:√50 is not yet in its simplified form since 50 has a factor that is a perfect square number.

Since we are dealing with square roots, we can think of a factor of 50 that is a perfect square and express 50 as a product of that number and another number.

Take note that 25 is a perfect square number and 25 x 2 = 50. Therefore, we can express √50 as

As per the second property of radicals (i.e., “the root of the product of given numbers is equal to the product of the roots of the given numbers”), we can express the answer above as √25 x √2.

We know that √25 = 5. Therefore, √25 x √2 = 5 x √2 or 5√2.

That’s it! We have simplified √50into5√2. Note that 5√2 has no perfect square factors anymore.

Question 21

Example 2:Simplify the following:

√27x³

Answer 21

Solution:The given radical is not in its simplest form since 27 still has a factor that is a perfect square (which is 9) and the exponent of its radicand (which is 3 in x3) is greater than the index (which is 2).

We know that 9 is a perfect square number and a factor of 27. Thus, we can express 27 as 9 ⋅ 3.

Meanwhile, we look for a factor ofx3that has the same power as the index. Our index is 2 so we look for a factor ofx3that has 2 as an exponent. In other words, we must factorx3in a manner that it has a factor with an exponent similar to the index (which is 2).x2is a factor ofx3sincex2⋅x = x3.

This means that we can factor the given radicand as follows:

Applying the second property of radicals, we can express the root of a product as a product of the roots.

Finally, we can apply property #1 which states that if the index and the exponent of the radical have the same value, then we can eliminate the radical sign and leave the radicand alone. Meanwhile, those radicals with radical signs that aren’t removed will be combined.

Therefore, the answer to our problem is 3x√3x.

Question 22

Example 3:Simplify the radical expression below

³√16y⁵

Answer 22

Solution:The given radical expression is not in simplified form since the number under the cube root sign has a factor that is a perfect cube and the exponent ofy5is greater than the index.

To simplify this expression, we think of a factor of 16 that is a perfect cube and express 16 as a product of that factor and another number. Note that 8 is a perfect cube number and 8 ⋅ 2 = 16.

Meanwhile, we can factory5with a factor that has an exponent equal to the index (which is 3). In particular,y5= y3⋅y2

Thus, we can express the given radical expression as follows:

Using property #2 of radicals, we can express the root of a product as the product of the roots:

Lastly, using property # 1, we can cancel the radical sign of those expressions with the same index and power of the radicand to come up with the final answer.

2y ³√2y²

Question 23

Example 4:Simplifythe radical expression below

√128a³b⁵

Answer 23

Solution:The given radical expression is not in simplified form since 128 has a factor that is a perfect square (which is 64) and the exponents of the radicand are greater than the index of the radical (which is 2).

Note that we can express 128 as 64 ⋅ 2.

On the other hand, we can expressa3as a product of two factors, one of which has an exponent equal to the index (which is 2). In particular,a2⋅a.

Meanwhile,b5can also be expressed as a product of three factors, two of which have an exponent of 2. In particular,b5= b2⋅b2⋅b

Therefore, we can express the given radical expression as:

Following the second property of radicals:

Lastly, as per the first property of radicals, we can remove the radical sign of those expressions with exponents equal to the index of the radical.

8ab²√2ab

Question 24

What Does “Rationalize the Denominator” Mean?

Answer 24

In simple words, rationalizing the denominator of a radical expression means removing the radical from the denominator of an expression.

For instance, 1⁄√x can be simplified by rationalizing its denominator. This means we can write it using its equivalent expression without a radical in the denominator. After performing this process, the expression 1⁄√x will be √x⁄x(we will learn the steps later). Note that the resulting expression has no radical in the denominator.

Question 25

How To Simplify Radical Expressions by Rationalizing the Denominator

Answer 25

To simplify radical expressions by rationalizing the denominator, follow these steps:

1. Multiply the numerator and the denominator by a certain radical that will remove the radical in the denominator.

Tip: If the radical in the denominator is a square root, then you can multiply the numerator and the denominator by a radical that will make the radicand a perfect square number. If the radical in the denominator is a cube root, then you can multiply the numerator and the denominator by a radical that will make the radicand a perfect cube number.

2. Simplify the result, if possible.

Question 26

Example 1:Simplify the following radical expression:

1/√2

Answer 26

Solution:The given expression has a radical in the denominator so we need to rationalize it.

1.Multiply the numerator and the denominator by a certain radical that will remove the radical in the denominator.

Our goal is to remove the radical in the denominator of the given expression (which is √2). If we multiply it by √2, we will obtain √4 (a perfect square number) resulting in the removal of the radical sign.

Thus, we can multiply both the numerator and the denominator by √2:

2.Simplify the result, if possible.

The result we have obtained which is √2⁄2 is already in simplified form. Therefore, the answer to our problem is √2⁄2.

Question 27

Example 2:Simplify the radical expression below by rationalizing its denominator.

3/√4x

Answer 27

Solution:

1.Multiply the numerator and the denominator by a certain radical that will remove the radical in the denominator.

Our goal is to remove the radical in the denominator (which is √4x).

Note that we can simplify the denominator √4x into 2√x. So, our focus now is to remove the radical sign of √x in 2√x. If we multiply it by √x, we will obtain 2√x2= 2x which enables us to remove the radical sign.

Hence, we can multiply both the numerator and the denominator by √x:

2.Simplify the result, if possible

The resulting expression we have obtained is 3√x⁄2x. This expression is already in simplified form so we can skip this step. Hence, the answer to the given problem is 3√x⁄2x.

Question 28

Example 3:Simplify the following radical expression:

5/√ab

Answer 28

Solution:The denominator is √ab. Notice that we can remove the radical sign of √ab if we make it √a2b2(having the same index and power of radicands removes the radical sign). Thus, we can multiply √ab by √ab so the result will be √a2b2or simplyab.

Hence, we should multiply both the numerator and the denominator of the given radical expression by √ab:

=5√ab/ab

Question 29

Example 4:Simplify the following radical expression:

2xy/x√y

Answer 29

Solution:The denominator of the given radical expression isx√y. We can remove the radical sign of √y by making the radicandya perfect square quantity. This means that we need to transformyintoy2.

This is possible by multiplying √y by √y. Since √y ⋅ √y = √y2= y .

So, we multiply both the numerator and the denominator by √y:

Hence, the answer is 2√y.

Question 30

an expression with the same terms as a given expression but with the opposite sign in the middle.

Answer 30

Conjugate

Question 31

Example:Determine the conjugate of the following:

a) 1 –√3

b)√7 +√3

Answer 31

a) 1 +√3

b)√7 -√3

Question 32

How To Rationalize the Denominator Using the Conjugate: 2 Steps

Answer 32

1. Multiply the numerator and the denominator by the conjugate of the denominator.
2. Simplify the result, if possible.

Question 33

Example 1:Simplifythe following radical expressionwhich we used as an example above:

1/√2+√5

Answer 33

Solution:

1.Multiply the numerator and the denominator by the conjugate of the denominator

The conjugate of √2 + √5 is √2 – √5.

2.Simplify the result, if possible

The answer we have obtained is already in the simplest form so it is our final answer.

=√2 - √5/-3

Question 34

Example 2:Simplify the radical expression below by rationalizing the denominator using the conjugate.

x/√a + √b

Answer 34

Solution:

=x√a - x√b/a -b

Question 35

How To Add and Subtract Radical Expressions

Answer 35

Here are the steps to add and subtract radical expressions:

1. Simplify the given radical expressions. If the given radical expressions are already in simplified form, skip this step.
2. Add or subtract the coefficients of the like radicals in the resulting expressions from step 1. You cannot add or subtract unlike radicals.

Let us have some examples:

Question 36

Example 1:Compute 3√a + 2√a – 4√a

Answer 36

Solution:

1.Simplify the given radical expressions. If the given radical expressions are already in simplified form, skip this step.

All radical expressions in the given problem are in their simplest form, so we can skip this step.

2.Add or subtract like radicals in the resulting expressions from step 1. You cannot add or subtract, unlike radicals.

Since all the radicals are like radicals, we can just add/subtract their coefficients and copy the common radical:

3√a + 2√a – 4√a

5√a – 4√a

√a

Hence, the answer is√a

Question 37

Example 2:Add√20 and√5

Answer 37

Solution:

1.Simplify the given radical expressions. If the given radical expressions are already in simplified form, skip this step.

We can simplify √20as 2√5. Thus, we have:

√20 and √5

2√5 + √5

2.Add or subtract like radicals in the resulting expressions from step 1. You cannot add or subtract, unlike radicals.

Since 2√5 and √5 are like radicals, we can combine them:

2√5 + √5 = 3√5

Therefore, the answer is3√5

Question 38

Example 3:
6ab√2b + √8a²b³+√a

Answer 38

Solution:

1.Simplify the given radical expressions. If the given radical expressions are already in simplified form, skip this step.

If we simplify each expression in the given, we have the following result (kindly review how to simplify radical expressions in the previous section):

2.Add or subtract like radicals in the resulting expressions from step 1. You cannot add or subtract, unlike radicals.

Based on what we have derived, we can only add like radicals 6ab√2b and 2ab√2b; we cannot combine √a to them.

Thus,

The answer is 8ab√2b + √a.

Question 39

Example 1:Multiply√2 by√4

Answer 39

Solution:Since both 2 and 4 have the same index (which is 2), then we can just multiply the radicands (2 and 4) and the coefficients (both are 1).

√2⋅ √4 = √8

√8 is the product we have obtained. However, it is not yet in its simplified form since it has a factor that is a perfect square (which is 4). So we need to simplify it:

√8 = √4 ⋅ √2 = 2√2

Hence, the final answer is2√2

Question 40

Example 2:Determine the product of 3√x and -2√xy

Answer 40

Solution:Since the given radicals have the same index (which is 2), we can just multiply the radicands and the coefficients.

We can simplify the product further:

Thus, the final answer should be-6x√y

Question 41

Example 3:Determine the product of the following:

√3xy and √2x²z

Answer 41

Solution:Since the given radicals have the same indices, then we can just multiply the radicands.

x√6xyz

Question 42

How To Make the Index of Two Radicals Similar.

Answer 42

Write the given radicals as expressions with fractional exponents. You will notice that the fractional exponents are dissimilar fractions.

Make the fractional exponents similar using their LCD. Write the expressions using the similar fractional exponents.

Rewrite the expressions with similar fractional exponents in radical form.

Question 43

Example 1:Make the indices of√3 and∛2 similar.

Answer 43

Solution:

1.Write the given radicals as expressions with fractional exponents. You will notice that the fractional exponents are dissimilar fractions.

Recall that to transform a radical into an expression with a fractional exponent, we write the index as the denominator of the fractional exponent and the power of the radicand as the numerator (kindly review this concept in our previous section above).

This means that √3 = 31/2and ∛2 = 21/3

Notice that the fractional exponents (which are ½ and ⅓ ) are dissimilar.

2.Make the fractional exponents similar using their LCD. Write the expressions using similar fractional exponents.

If we make ½ and ⅓ similar, we will obtain 3/6 and 2/6.

Hence, we have 33/6and 22/6

3.Rewrite the expressions with similar fractional exponents in radical form

That’s it! We’ve been able to convert the given radicals with different indices into equivalent radicals with similar indices.

Question 44

How To Multiply Radicals With Different Indices.

Answer 44

To multiply radicals with different indices, you have to first make the indices similar. Once you’ve converted them into radicals with the same index, follow the steps on multiplying radicals with similar indices.

Question 45

Example 1:Multiply√x by∛x

Answer 45

Solution:

The given radicals have different indices so we have to make them similar first. Let us start by writing the given radicals into an expression with fractional exponents, then make the fractional exponents similar.

Converting the expressions with fractional exponents into radical form, we’ll obtain:

Since the radicals now have the same index, we can just multiply the radicands:

The answer can’t be simplified further so it should be the final answer.

=⁶√x⁵

Question 46

Example 1:Divide√10 by√2

Answer 46

Solution:We can write the given problem as:

Invoking the third property of radicals allows us to write the problem as:

We can now divide the radicands:

Hence, the answer is√5

Question 47

Example 2:Divide√18 by√5

Answer 47

Solution:

We can write the given problem as:

Note that it is not advisable to use the third property this time since 18 is not divisible by 5. Instead, we can just simplify the expression by rationalizIng the denominator.

To rationalize the denominator, we can multiply both the numerator and the denominator by √5so that the denominator will be √25 which is a perfect square number.

Simplifying the result:

3√10/5

Question 48

To solve a radical equation, follow these steps:

Answer 48

Isolate the terms that are under the radical sign from the terms that are not under the radical sign. This means that only one side of the equality must contain the terms under the radical sign.Raise both sides of the equation by the power equivalent to the index of the radical to remove the radical sign.Solve the resulting linear/quadratic equation.

Question 49

Example 1:Solve for the value of x in the equation
√x + 3 = 12

Answer 49

Solution:

1.Isolate the terms that are under the radical sign from the terms that are not under the radical sign. This means that only one side of the equation must contain the terms under the radical sign.

Looking at √x + 3 = 12, we must isolate x from the other quantity on the left-hand side. This can be achieved if we transpose 3 to the right-hand side of the equation.

√x + 3 = 12

√x = -3 + 12

√x = 9

2.Raise both sides of the equation by the power equivalent to the index of the radical to remove the radical sign.

The index of the radical in the equation is 2. Thus, we need to raise both sides of the equation by 2 to remove the radical sign.

√x = 9

(√x)2= (9)2

x = 81

3.Solve the resulting linear/quadratic equation.

The resulting equation is justx = 81which tells us that the solution of the equation is 81.

Thus, the answer to the radical equation isx = 81.

Question 50

Example 2:Solve for x in √x²+19=10

Answer 50

Solution:

1.Isolate the terms that are under the radical sign from the terms that are not under the radical sign. This means that only one side of the equation must contain the terms under the radical sign.

The terms under the radical sign in the given problem is already isolated. So, we can skip this step.

2.Raise both sides of the equation by the power equivalent to the index of the radical to remove the radical sign.

The index of the radical is 2 so we need to raise both sides of the equation by 2 to remove the radical sign.

3. Solve the resulting linear/quadratic equation.

The resulting equation isx2+ 19 = 100. This equation is a quadratic equation.

Let us solve the equation:

x2+ 19 = 100

x2= -19 + 100transposition method

x2= 81

√x2=√81 extracting the square root of both sides

x = ± 9

This means that the solutions of the radical equations are9 and -9

Question 51

1) Simplify ³√x⁴y⁵

a) xy³√𝑥𝑦²
b) x³y² ³√𝑥𝑦²
c)³√𝑥³𝑦⁴
d) xy³√𝑥²𝑦³

Answer 51

A

Question 52

2) Which of the following is not equivalent to √32?

a) 32^1/2
b) 32^2/4
c) 4(2)^½
d) 16(2)^½

Answer 52

D

Question 53

3) Add: 4√5 and 7√5

a) √55
b) 11√5
c) 28√5
d)√5

Answer 53

B

Question 54

4) Solve for the value of x in √𝑥 − 2 = 5

a) 25
b) 26
c) 27
d) 28

Answer 54

C

Question 55

5) Rationalize the denominator of 1/2√x

a)𝑥/2√𝑥
b)√𝑥/2𝑥
c)2/√𝑥
d) None of the above

Answer 55

B