Adding and Subtracting Radicals: Videos & Practice Problems

Adding and subtracting radicals requires combining like radicals, which have the same radicand and index. Simplify radicals by factoring out perfect powers to convert unlike radicals into like radicals before combining. For example, $\sqrt{20}$ simplifies to $2\sqrt{5}$, allowing addition with $\sqrt{5}$. Remember, you cannot combine radicals with different indexes or add radicals by merging radicands, such as $\sqrt{7}+\sqrt{7}\backslash neq\sqrt{14}$. This understanding is essential for simplifying polynomial expressions involving radicals.

Adding & Subtracting LIKE Radicals

Understanding how to add and subtract radical expressions is essential in algebra, and it closely resembles the process of combining algebraic terms. When simplifying expressions with radicals, the key is to identify and combine like radicals, which share the same radicand (the number or expression inside the radical) and the same index (the root type, such as square root or cube root).

For instance, if you have an expression like \(2\sqrt{x} + 4\sqrt{x}\), you can combine these terms because they both contain the same radicand \(x\) and are both square roots. This results in \(6\sqrt{x}\). Similarly, constants can be combined in the same way, such as \(3 + 8 = 11\).

However, caution is necessary when dealing with different types of radicals. For example, in the expression \(3\sqrt{7} + 2\sqrt{7} - \sqrt[3]{7}\), the first two terms can be combined to yield \(5\sqrt{7}\) since they share the same radicand and index. The term \(-\sqrt[3]{7}\) cannot be combined with the others because it has a different index (cube root versus square root), making it a different type of radical.

It is also important to note that you cannot merge separate radicals into a single radical. For example, \(\sqrt{7} + \sqrt{7}\) does not equal \(\sqrt{14}\); instead, it simplifies to \(2\sqrt{7}\). This common mistake can lead to incorrect answers, so always ensure that you are only combining like terms.

In another example, consider \(9\sqrt[3]{x} + 4 - \sqrt{x}\). Here, \(9\sqrt[3]{x}\) and \(\sqrt{x}\) cannot be combined due to their differing indexes, even though they share the same radicand \(x\). You can combine the constant \(4\) with any other constants present, but the radical terms remain separate.

In summary, when working with radical expressions, focus on identifying like radicals based on their radicands and indexes, combine them accordingly, and avoid merging different types of radicals to ensure accurate simplification.

True or False:
$\sqrt{9+16}$ and $\sqrt{9}+\sqrt{16}$ are equal.

• A. True
• B. False
• C. Cannot be determined

Answer: (B)

Add or subtract as indicated and simplify.
$2\sqrt{5}+6\sqrt{5}$

• A. $12\sqrt5$
• B. $4\sqrt5$
• C. $8\sqrt5$
• D. $8\sqrt{10}$

Answer: (C)

Add or subtract as indicated and simplify.
$6\sqrt[4]{3y}+2\sqrt[4]{3y}-5\sqrt[4]{3y}$

• A. $3\sqrt[4]{3y}$
• B. $5\sqrt[4]{3y}$
• C. $\sqrt[4]{3y}$
• D. $3\sqrt[4]{6y}$

Answer: (A)

Add or subtract as indicated and simplify.
$\sqrt[3]{50}+\sqrt[3]{8}-\sqrt[3]{18}$

• A. $\sqrt{40}$
• B. $\sqrt[3]{40}$
• C. $\sqrt[3]{40}+2$
• D. Not like a radical

Answer: (D)

Adding & Subtracting UNLIKE Radicals

When working with radicals, it's essential to understand how to add and subtract them, especially when they are not like radicals. Like radicals have the same radicand (the number under the square root) and index, allowing for straightforward addition or subtraction. For instance, combining \(3\sqrt{5}\) and \(4\sqrt{5}\) results in \(7\sqrt{5}\) because the radicands are the same.

However, when faced with unlike radicals, such as \(\sqrt{5}\) and \(\sqrt{20}\), the first step is to simplify them. Simplification involves breaking down the radicals into their prime factors to identify any perfect squares. In the case of \(\sqrt{20}\), it can be expressed as \(\sqrt{4 \cdot 5}\), which simplifies to \(2\sqrt{5}\). This transformation allows us to rewrite the original expression as \(\sqrt{5} + 2\sqrt{5}\), which can then be combined to yield \(3\sqrt{5}\).

Another example involves \(5\sqrt{2}\) and \(\sqrt{18}\). Here, \(\sqrt{18}\) can be simplified to \(\sqrt{9 \cdot 2}\), resulting in \(3\sqrt{2}\). Thus, the expression becomes \(5\sqrt{2} - 3\sqrt{2}\), which simplifies to \(2\sqrt{2}\).

In cases where both radicals need simplification, such as \(\sqrt{18}\) and \(\sqrt{50}\), we can break them down as follows: \(\sqrt{18} = 3\sqrt{2}\) and \(\sqrt{50} = 5\sqrt{2}\). This allows us to combine them into \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\).

In summary, the key to adding or subtracting unlike radicals is to simplify them first, transforming them into like radicals before performing the operation. This method not only clarifies the process but also ensures accuracy in calculations.

Add or subtract as indicated and simplify.
$\sqrt{12}+\sqrt{20}-\sqrt{45}$

• A. $2\sqrt3-\sqrt5$
• B. $\sqrt3-2\sqrt5$
• C. $4\sqrt3+\sqrt5$
• D. $4\sqrt5+\sqrt3$

Answer: (A)

Adding & Subtracting UNLIKE Radicals Example 1

When simplifying expressions involving roots with different indices, it's important to first identify any perfect powers within the radicals. For example, the cube root of 125 simplifies because 125 is a perfect cube, as \$5^3 = 125\(. Similarly, the fourth root of 81 simplifies since \)3^4 = 81\(. This allows the expression \)\sqrt{50} + \sqrt[3]{125} - \sqrt[4]{81}\( to be rewritten as \)\sqrt{50} + 5 - 3\(. Combining the constants \)5 - 3\( results in \)2\(, simplifying the expression to \)\sqrt{50} + 2\(.

Further simplification of the square root term involves breaking down 50 into a product of a perfect square and another factor. Using the product rule for square roots, \)\sqrt{50}\( can be expressed as \)\sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}\(. Since \)\sqrt{25} = 5\(, this simplifies to \)5\sqrt{2}\(. Therefore, the fully simplified expression is \)5\sqrt{2} + 2$.

This process highlights the importance of recognizing perfect powers and applying root properties to simplify radical expressions effectively. By decomposing radicals into factors containing perfect powers, you can transform complex roots into simpler terms, making it easier to combine and evaluate expressions.

Adding & Subtracting UNLIKE Radicals Example 2

When simplifying expressions involving radicals with variables, it helps to recognize perfect powers and apply root properties systematically. Consider the expression involving the square root of 32x, the cube root of 8x³, and the fourth root of 16x⁴. First, identify perfect powers within each term. The cube root of 8x³ simplifies because 8 is a perfect cube (2³), and the cube root of x³ is simply x. Thus, the cube root of 8x³ becomes 2x.

Next, the fourth root of 16x⁴ can be simplified by noting that 16 is 2⁴, so the fourth root of 16 is 2. Similarly, the fourth root of x⁴ is x. Therefore, the fourth root of 16x⁴ simplifies to 2x. Since this term is subtracted, it becomes -2x.

Now, observe that the terms 2x and -2x cancel each other out, leaving only the square root of 32x to simplify. To simplify the square root of 32x, factor 32 into 16 × 2, where 16 is a perfect square. Using the property of radicals, the square root of a product equals the product of the square roots: \(\sqrt{32x} = \sqrt{16} \times \sqrt{2x}\). Since \(\sqrt{16} = 4\), the expression simplifies to \$4\sqrt{2x}\(.

Thus, the original complex radical expression simplifies neatly to \)4\sqrt{2x}$. This process highlights the importance of recognizing perfect powers and applying root and exponent rules to simplify radical expressions involving variables effectively.

Here’s what students ask on this topic:

How do you add and subtract radicals with the same radicand and index?

To add or subtract radicals, they must be like radicals, meaning they have the same radicand (the number inside the radical) and the same index (such as square root or cube root). When this condition is met, you simply add or subtract the coefficients (the numbers in front of the radicals) just like combining like terms in algebra. For example, $3\sqrt{7}+2\sqrt{7}$ becomes $5\sqrt{7}$. Radicals with different radicands or indexes cannot be combined directly.

What should you do when adding radicals that are not like radicals?

When radicals are not like radicals (different radicands or indexes), you first simplify each radical by factoring out perfect powers. This process can convert unlike radicals into like radicals. For example, to add $\sqrt{5}+\sqrt{20}$, simplify $\sqrt{20}$ as $\sqrt{4}\sqrt{5}$ which equals $2\sqrt{5}$. Now you have $\sqrt{5}+2\sqrt{5}$, which are like radicals and can be combined to $3\sqrt{5}$.

Can you add radicals with different indexes, like a square root and a cube root?

No, you cannot add radicals with different indexes directly. For example, $\sqrt{7}+\sqrt[3]{7}$ cannot be combined because the first is a square root (index 2) and the second is a cube root (index 3). They are not like radicals. You must keep them separate or simplify further if possible, but generally, different indexes mean the radicals cannot be added or subtracted.

Why can't you combine radicals by adding their radicands, like $\sqrt{7}+\sqrt{7}=\sqrt{14}$?

You cannot combine radicals by adding their radicands because the square root of a sum is not equal to the sum of the square roots. For example, $\sqrt{7}+\sqrt{7}$ equals $2\sqrt{7}$, not $\sqrt{14}$. This is a common mistake. Radicals can only be combined by adding or subtracting their coefficients when the radicals themselves are identical.

How do you simplify expressions like $\sqrt{18}+\sqrt{50}$ before adding?

To simplify $\sqrt{18}+\sqrt{50}$, factor each radicand to pull out perfect squares. For $\sqrt{18}$, factor as $\sqrt{9*2}$ which simplifies to $3\sqrt{2}$. For $\sqrt{50}$, factor as $\sqrt{25*2}$ which simplifies to $5\sqrt{2}$. Now the expression is $3\sqrt{2}+5\sqrt{2}$, which can be combined to $8\sqrt{2}$.