What are radical expressions?
Radical expressions (or radicals) are mathematical expressions that contain a radical (√) sign. The expression $\sqrt[n]{x}$ is read as “x radical n,” or “the nth root of x.” When finding the nth radical or nth root, you want to get the expression that when raised to the nth power, you would get the same expression inside the radical sign. The root of a mathematical expression is the reverse of raising it to power. For example, if you want to solve the square root of an expression, you need to find another expression, such that when you square it (or raise it to the power of 2), you get what is inside the radical sign.
There are three important components of any radical expression. These are the radical sign, the radicand, and the index.
The radical sign (√) is a symbol used to indicate a root. The radicand is the numerical expression or algebraic expression within the radical sign. The index is the small number placed above the “v” portion of the radical sign. In the radical expression above, we can say that the radicand is 64, and the index is 3.
When there is no index number written, it is understood that the index is 2 or square root. So, the radical expression $\sqrt{x}$ is read as “the square root of x.” If there is an index number other than the number 2, you need to find a root other than a square root.
Like and unlike radicals
In adding or subtracting radical expressions, it is crucial to have a knowledge of like and unlike radicals. It is because the operation of addition and subtraction radical expressions involves combining radicals which can only be performed on like radicals. Combining radicals is feasible when the index and the radicand of two or more radicals are the same. If the radicand and index of two or more radicals are the same, they are known as like radicals. It is useful to treat radicals like variables because like radicals can be combines in the same way that like variables can combined. Sometimes, you need to simplify the radicands in a radical expression before adding or subtracting like radicals.
For example, $2\sqrt{2}$ and $3\sqrt{2}$ is like radicals because the radicands and the index are the same. On the other hand, $2\sqrt[3]{2}$ and $3\sqrt[5]{2}$ are “unlike radicals” because their indices are different even though they have the same radicand.
To combine like radicals, simply add the numerical coefficients and just copy the radical part. For example, the expression $2\sqrt{2}$ and $3\sqrt{2}$ are like radicals so they can be combined. To combine, add the numerical coefficients 2 and 3, that is, 2 + 3, then attach the radical $\sqrt{2}$. Thus, $2\sqrt{2}$ and $3\sqrt{2}$ combined are $5\sqrt{2}$.
On the other hand, we cannot combine unlike radicals into a single term. For example, the radical $2\sqrt[3]{2}$ and $3\sqrt[5]{2}$ are unlike radicals because they have different indices. So, combining them will result in an expression with two terms $2\sqrt[3]{2}$ and $3\sqrt[5]{2}$.
How to simplify radical expressions?
Just like in simplifying other mathematical expressions, we can simplify radicals by combining “like terms” or “like radicals” through the process of addition or subtraction.
There are two essential processes to take note of when combining radicals by addition or subtraction. The first one is to look at the index, and the second is to look at the radicand. If the given radicals have the same radicand and index, then addition and subtraction are feasible. If not, then you cannot combine the two radicals. This is similar to combining like terms in a polynomial expression.
You can treat the radicals as variables and combine like terms by adding or subtracting their numerical coefficients and attaching their common variable. For example, to combine $\sqrt{xy}$ and $3\sqrt{xy}$, we can have $\sqrt{xy} + 3\sqrt{xy}$ . The radical $\sqrt{xy}$ has a numerical coefficient of 1 so we can visualize it as $1\sqrt{xy}$ while the radical $3\sqrt{xy}$ has a numerical coefficient of 3. To combine, simply add their numerical coefficients. So, $\sqrt{xy}+ 3\sqrt{xy}=(1+3)\sqrt{xy}=4\sqrt{xy}$.
Simplifying radical expressions is the process of writing them in the most efficient and compact structure possible while maintaining the value of the original expression. It is a valuable mathematical skill because it converts complex or difficult-to-read expressions into simpler ones. Below are some rules and steps in simplifying radical expressions.
- Find the prime factors of the radicand. Ignore the index for now and just look at radicand. Factor the number in the radicand by writing it as the product of two smaller numbers.
Example: Simplify $\sqrt{63}$.
Find some factors of 63. You can’t divide 63 by 2, so try dividing it by 3 instead: 63 ÷3=21, so 63 =3 ×21. We can rewrite the radical expression $\sqrt{63}$ as $\sqrt{3\:x\:21}$.
Repeat the process until the radicand is factored in completely. Remember that any number can be factored down into prime numbers such as 2, 3, 5, 7, and so on.
We now have $\sqrt{3\:x\:21}$ but we can still factor 21 as 3 ×7. Now we can rewrite $\sqrt{63}$ as $\sqrt{3\:x\:3\:x\:7}$.
- Rewrite groups of the same prime factors in exponent form. If the same prime factor appears more than once, rewrite them in exponent form. For example, if factor 3 appears five times, you can rewrite it as 35. Since we are looking for the square root, the same factors can be written as as powers of 2. Thus, we can rewrite the expression $\sqrt{3\:x\:3\:x\:7}$ as $\sqrt{3^{2}\:x\:7}$.
- Simplify the root of exponents whenever possible. Finding the roots and raising numbers to exponents are opposite processes, so they cancel each other out. For example, if we get the square root x2, written as $\sqrt{x}$, the exponent of x and the radical sign will be canceled out. Hence, the square root of x2 is simply x or $\sqrt{x}=x$ So, if any factors of $\sqrt{63}$ are raised to the power of 2, move the factor in front of the square root and remove the exponent. We can write $\sqrt{3^{2}\:×\:7}\:as\:\sqrt{3^{2}}\:\sqrt{7}$ if the radicand is expressed as one multiplication problem. Thus, $\sqrt{3^2}\:\sqrt{7}\:=\:3\sqrt{7}$.
- Simplify any multiplication and exponents. You might encounter multiple numbers in front of the radical sign or underneath it in more complicated radical expression problems. Solve the multiplication problems outside and inside the radical sign to simplify the answer.
For example, simplify $\sqrt[5]{2^{5}\:×\:3^{7}\:×5}$. This expression is already factored into prime numbers, so we can skip the factorization step. Rewrite the expression as $\sqrt[5]{2^{5}}\sqrt[5]{3^{7}}\sqrt[5]{5}$. In $\sqrt[5]{2^{5}}$, the root and exponent can be canceled out to make 2. The $\sqrt[5]{3^{7}}$ is equal to $\sqrt[5]{3^{5}\:×\:3^{2}}$ By the rules of exponents. Canceling the root and exponent out will make it $3\sqrt[5]{3^{2}}$. The $\sqrt[5]{5}$ has no exponents that cancel out, so this cannot be further simplified. Thus, $\sqrt[5]{2^{5}\:x\:3^{7}\:x\:5}\:=\:2\:x\:3\:\sqrt[5]{3^{2}}\:x\:\sqrt[5]{5}$. Combining like radicals will give us $(2\:x\:3)\:\sqrt[5]{3^{2}\:x\:5}$. Simplifying further, $\sqrt[5]{2^{5}\:x\:3^{7}\:x\:5}$ is equal to $6\sqrt[5]{45}$
How to add radical expressions?
In adding radical expressions, we combine the like radicals, and the unlike radicals are written as they are. When combining like radicals, make sure to group them and keep the signs of all the terms in the radical expression the same.
Example #1
Solve for $3\sqrt{2}\:+\:2\sqrt{2}$.
Solution
| Radicals Addition Process | Step-by-step explanation |
| $3\sqrt{2}\:+\:2\sqrt{2}$ | Set-up addition. |
| $5\sqrt{2}$ | Combine like radicals |
| Therefore, $3\sqrt{2}\:+\:2\sqrt{2}\:is\:5\sqrt{2}$. |
Example #2
Solve for $6y\sqrt{2x^5}\:-\:2y\sqrt{2x^5}$.
Solution
| Radicals Addition Process | Step-by-step explanation |
| $6y\sqrt{2x^5}\:-\:2y\sqrt{2x^5}$ | Set-up addition. Notice that we can get the square root of x5. |
| $6x^{2}y\sqrt{2x}\:-\:2x^{2}y\sqrt{2x}$ | Solve for the square root of x5. |
| $4x^{2}y\sqrt{2x}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, $6y\sqrt{2x^5}\:-\:2y\sqrt{2x^5}$ is equal to $4x^{2}y\sqrt{2x}$. |
Example #3
Find the sum of $-6\sqrt[3]{2b^2}\:-\:\sqrt{2b^2}\:+\:7\sqrt[3]{2a^2}\:and\:11\sqrt[3]{2b^2}\:-\:2\sqrt{2b^2}\:+\:8\sqrt[3]{16a^2}$.
Solution
| Radicals Addition Process | Step-by-step explanation |
| $(-6\sqrt[3]{2b^2}\:-\:\sqrt{2b^2}\:+\:7\sqrt[3]{2a^2})\:+(\:11\sqrt[3]{2b^2}\:-\:2\sqrt{2b^2}\:+\:8\sqrt[3]{16a^2})$ | Set-up addition. |
| $(-6\sqrt[3]{2b^2}\:-\:b\sqrt{2}\:+\:7\sqrt[3]{2a^2})\:+(\:11\sqrt[3]{2b^2}\:-\:2b\sqrt{2}\:+\:16\sqrt[3]{2a^2})$ | Simplify the radicals in each term of the expression. Notice that the radical $\sqrt{2b^2}$ can be simplified as $b\sqrt{2}$. The term $8\sqrt[3]{16a^2}$ can be simplified as $16\sqrt[3]{2a^2}$ |
| $-6\sqrt[3]{2b^2}\:+\:11\sqrt[3]{2b^2}\:-b\sqrt{2}\:-2b\sqrt{2}+\:7\sqrt[3]{2a^2}\:+\:16\sqrt[3]{2a^2}$ | Remove the grouping symbols. The sign of the terms in the expression remains the same. Arrange the terms of the new expression such that like radicals are beside each other. |
| $5\sqrt[3]{2b^2}\:+\:b\sqrt{2}\:+\:23\sqrt[3]{2a^2}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the sum of $ -6\sqrt[3]{2b^2}\:-\:\sqrt{2b^2}\:+\:7\sqrt[3]{2a^2}\:and\:11\sqrt[3]{2b^2}\:-\:2\sqrt{2b^2}\:+\:8\sqrt[3]{16a^2} $ is $5\sqrt[3]{2b^2}\:+\:b\sqrt{2}\:+\:23\sqrt[3]{2a^2}$. |
How to subtract radical expressions?
The process for subtracting radical expressions is the same as the process for adding them. The only difference is that when subtracting one radical expression from another, you must change the signs of each term in the expression being subtracted and then combine like radicals.
But why do we have to change the signs of a mathematical expression when being subject to subtraction? Let us take an example of two numbers, 7 and −2. Suppose if we must subtract −2 from 7, we write it as 7−(−2). We know that the product of two positive signs or two negative signs is positive, and the product of two unlike signs is negative. In the example: 7−(−2) = 7 + 2 = 9. If we do not change the sign, we will have 7 – 2 = 5, which is a completely different result. So, changing the signs of the subtrahend when subtracting expressions is necessary. In other words, the subtraction symbol (−) must be distributed to each term of the subtrahends before combining mathematical terms.
Example #1
Find the difference between $3\sqrt{3}$ and $-\:2\sqrt{3}$?
Solution
| Radicals Subtraction Process | Step-by-step explanation |
| $3\sqrt{3}-(-\:2\sqrt{3})$ | Set-up subtraction. |
| $3\sqrt{3}\:+\:2\sqrt{3}$ | Distribute the minus sign to the term in the subtrahend. Thus, the $-\:2\sqrt{3}$ will become $2\sqrt{3}$. Then, proceed like in the process of addition. |
| $5\sqrt{3}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the difference between $3\sqrt{3}$ and $-\:2\sqrt{3}$ is $5\sqrt{3}$. |
Example #2
Subtract $3a\sqrt[3]{2b^5}$ from $7a\sqrt[3]{2b^5}$.
Solution
| Radicals Addition Process | Step-by-step explanation |
| $7a\sqrt[3]{2b^5}\:-\:(3a\sqrt[3]{2b^5})$ | Set-up subtraction. |
| $7a\sqrt[3]{2b^5}\:-\:3a\sqrt[3]{2b^5}$ | Distribute the minus sign to the term in the subtrahend. Thus, the $3a\sqrt[3]{2b^5}$ will become $-3a\sqrt[3]{2b^5}$. Then, proceed like in the process of addition. |
| $7ab\sqrt[3]{2b^2}\:-\:3ab\sqrt[3]{2b^2}$ | Notice that the radical $\sqrt[3]{2b^5}$ can be simplified as $b\sqrt[3]{2b^2}$. So, $7a\sqrt[3]{2b^5}$ can be simplified as $7ab\sqrt[3]{2b^2}$ while $-3a\sqrt[3]{2b^5}$ can be simplified as $-3ab\sqrt[3]{2b^2}$ |
| $4ab\sqrt[3]{2b^2}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the result of subtracting $3a\sqrt[3]{2b^5}$ from $7a\sqrt[3]{2b^5}$ is $4ab\sqrt[3]{2b^2}$. |
Example #3
Solve for the difference between $x\sqrt{18y^{2}}\:-\:\sqrt[3]{2x^3}\:+\:y\sqrt{18x^2}$ and $-\:2\sqrt{2y^2}\:+\:\sqrt[3]{54}$.
Solution
| Radicals Subtraction Process | Step-by-step explanation |
| $(x\sqrt{18y^{2}}\:-\:\sqrt[3]{2x^3}\:+\:y\sqrt{18x^2})\:-\:(-\:2\sqrt{2y^2}\:+\:\sqrt[3]{54})$ | Set-up subtraction. |
| $(3xy\sqrt{2}\:-\:x\:\sqrt[3]{2}\:+\:3xy\sqrt{2})\:-\:(-\:2y\sqrt{2}\:+\:3\sqrt[3]{2})$ | Simplify the radicals in each term of the expression. |
| $(6xy\sqrt{2}\:-\:x\sqrt[3]{2})\:-\:(-\:2y\sqrt{2}\:+\:3\sqrt[3]{2})$ | Combine the like radicals in each expression before proceeding to subtraction. |
| $6xy\sqrt{2}\:-\:x\sqrt[3]{2}\:+\:2y\sqrt{2}\:-\:3\sqrt[3]{2}$ | Distribute the minus sign to the term in the subtrahend. Thus, the $- 2y\sqrt{2}$ will become $2y\sqrt{2}$ and $3\sqrt[3]{2}$ will become $-3\sqrt[3]{2}$. Then, proceed like in the process of addition. |
| $6xy\sqrt{2}\:+\:2y\sqrt{2}\:-\:x\sqrt[3]{2}\:-\:3\sqrt[3]{2}$ | Arrange the terms of the new expression such that like radicals are beside each other. |
| $(6xy\:+\:2y)\sqrt{2}\:+\:(-x\:-3)\:\sqrt[3]{2}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the difference between $x\sqrt{18y^{2}}\:-\:\sqrt[3]{2x^3}\:+\:y\sqrt{18x^2}$ and $-\:2\sqrt{2y^2}\:+\:\sqrt[3]{54}$ is $(6xy\:+\:2y)\sqrt{2}\:+\:(-x\:-3)\:\sqrt[3]{2}$ |
How to solve problems involving adding and subtracting radical expressions?
To solve problems involving radical expressions, follow these steps:
- Analyze the problem.
- List all the given information.
- Set up the addition or subtraction process.
Problem 1
Cindy and Daisy both have a cubic terrarium. The volume of Cindy’s terrarium is 27x +54 cubic inches, while Daisy’s measures 8x+16. What is the difference between the length of the sides of their terrarium?
Solution
| Process | Step-by-step explanation |
| $T= \sqrt[3]{C}\:-\:\sqrt[3]{D}$ | Set up the working formula. To get the length of the sides of a cube, we must get the cube root of its volume. Let T be the difference of the length of the sides of the terrariums, let C be the volume of Cindy’s terrarium, and let D be the volume of Daisy’s terrarium. |
| $T= \sqrt[3]{27x\:+\:54}\:-\:\sqrt[3]{8x\:+\:16}$ | Plugin the values into the working formula. |
| $T = \sqrt[3]{27(x + 2)}\:-\: \sqrt[3]{8 (x + 2)}$ | Factor the radicands if possible. In this case, we can factor 27x +54 as 27(x + 2) and 8x+16 as 8(x+2) |
| $T = 3\sqrt[3]{x + 2}- 2\sqrt[3]{x + 2}$ | Get the square root of 27 and 8, so it can get out of the radical sign. So, $\sqrt[3]{27(x + 2)}- \sqrt[3]{8 (x + 2)} = 3\sqrt[3]{x + 2}- 2\sqrt[3]{x + 2}$ |
| $T = \sqrt[3]{x + 2}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the difference between the length of the sides of Cindy’s and Daisy’s terrarium is $\sqrt[3]{x +2}$ inches. |
Problem #2
Find the perimeter of a rectangle having a length of $\sqrt{x+1}$ inches and a width of $\sqrt{4x+4}$ inches.
Solution
| Process | Step-by-step explanation |
| P = 2L+2W. | Set- up the formula for the perimeter of a rectangle. The perimeter of a rectangle can be solved by adding the length of all the sides. Let P be the perimeter, let L be the length, and let W be the width. |
| $P= 2(\sqrt{x+1}) + 2(\sqrt{4x + 4})$ | Plugin the values to the working formula |
| $P = 2\sqrt{x+1} + 2\sqrt{4x + 4}$ | Remove the grouping symbols. |
| $P = 2\sqrt{x+1} + 2\sqrt{4(x + 1)}$ | Factor the radicands whenever possible. In this case, we can factor 4x+4 as 4(x+1) |
| $P = 2\sqrt{x+1} + 4\sqrt{(x + 1)}$ | Get the square root of 4, so it can get out of the radical sign. So, $2\sqrt{4(x + 1)} = 2(2)\sqrt{x + 1} = 4\sqrt{(x + 1)}$ |
| $P = 6\sqrt{x+1}$ | Combine the like radicals into a single term by adding their numerical coefficients. |
| Therefore, the perimeter of the rectangle is $6\sqrt{x+1}$ inches. |
Problem #3
A company released two new models of cellphones last month. If the sale of the first model is already reached $2y + \sqrt{4y^2 – 20}$ million dollars and the sale of the second model reached $3y + \sqrt{9y^2 – 45}$ million dollars, how much is the total sales of the company for the two new cellphone models in a month?
Solution
| Process | Step-by-step explanation |
| T = A + B | Set- up the formula for the perimeter of a rectangle. The total sales of the company can be solved by adding the sales of the two phone models. Let T be the total sales, let A be the sales of the first phone model, and let B be the sale of the second model. |
| $T=(2y+\sqrt{4y^3-20y^2})+(3y+\sqrt{9y^3-45y^2})$ | Plugin the values to the working formula |
| $T=(2y+\sqrt{4y^2(y-5))}+(3y+\sqrt{9y^2(y-5)}$ | Factor the radicands if possible. In this case, we can factor 4y3 – 20y2 as 4y2 (y – 5) and 9y3 – 45y2 as 9y2(y – 5) |
| $T=(2y+2y\sqrt{y-5})+(3y+3y\sqrt{y-5})$ | Get the square root of 4y2 and 9y2 , so they can be moved in front of the radical sign. So, $\sqrt{4y^{2}(y-5)}$ will be $2y\sqrt{y-5}$ and $\sqrt{9y^2(y-5)}$ will be $3y\sqrt{y-5}$ |
| $T=2y+2y\sqrt{y-5}+3y+3y\sqrt{y-5}$ | Remove the grouping symbols. The sign of the radical terms remains the same. |
| $T=2y+3y+2y\sqrt{y-5}+3y\sqrt{y-5}$ | Arrange the radical terms such that like radicals and like terms are beside each other. |
| $T=5y+5y\sqrt{y-5}$ | Combine the like radicals and terms into a single term by adding their numerical coefficients. |
| Therefore, the total sales of the company for the two new cellphone models in a month is $5y+5y\sqrt{y-5}$ million dollars. |
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