Simplifying Radical Expressions: Videos & Practice Problems

Understanding how to simplify radicals is essential in mathematics. To simplify expressions like $\sqrt{20}$, break them into products where one factor is a perfect power, such as $\sqrt{4}$ and $\sqrt{5}$. This leads to $2\sqrt{5}$. When dealing with fractions, apply similar principles to simplify both the numerator and denominator separately. Combining like radicals requires matching radicands and indices, ensuring efficient calculations.

Expanding Radicals

Understanding how to simplify square roots and cube roots is essential in mathematics, especially when dealing with non-perfect powers. A perfect square, like the square root of 9, equals 3, while a perfect cube, such as the cube root of 8, equals 2. However, many expressions, like the square root of 20 or the cube root of 54, require simplification. The key to simplifying these radicals is to express them as a product of two factors, where one factor is a perfect power.

For instance, to simplify the square root of 20, we can factor it into 4 and 5, since \(4 \times 5 = 20\). This allows us to separate the radical: \(\sqrt{20} = \sqrt{4} \times \sqrt{5}\). The square root of 4 is 2, while the square root of 5 remains as is, leading to the simplified expression \(2\sqrt{5}\). Since 5 is a prime number, we cannot simplify further, confirming that \(2\sqrt{5}\) is fully simplified.

When dealing with variables, the same principle applies. For example, to simplify \(\sqrt{18x^2}\), we can express 18 as \(9 \times 2\) and recognize that \(x^2\) is a perfect square. Thus, we have \(\sqrt{18x^2} = \sqrt{9} \times \sqrt{2} \times \sqrt{x^2}\). The square root of 9 is 3, and the square root of \(x^2\) is \(x\), leading to the expression \(3x\sqrt{2}\), which is fully simplified.

In the case of cube roots, the process is similar. For example, to simplify \(\sqrt[3]{54x^4}\), we can factor 54 into \(27 \times 2\) and express \(x^4\) as \(x^3 \times x\). This gives us \(\sqrt[3]{54x^4} = \sqrt[3]{27} \times \sqrt[3]{2} \times \sqrt[3]{x^3} \times \sqrt[3]{x}\). The cube root of 27 is 3, and the cube root of \(x^3\) is \(x\), resulting in \(3x\sqrt[3]{2x}\). This expression is also fully simplified.

In summary, the process of simplifying radicals involves breaking down the expression into factors, identifying perfect powers, and then simplifying each component. This method not only applies to numbers but also extends to variables, ensuring a comprehensive understanding of radical simplification.

Simplify the radical.
$\sqrt{75}$

• A. $3\sqrt5$
• B. $3\sqrt{25}$
• C. $5\sqrt3$
• D. $25\sqrt3$

Answer: (C)

Simplify the radical.
$\sqrt{180}$

• A. $6\sqrt5$
• B. $3\sqrt5$
• C. $3\sqrt{20}$
• D. $2\sqrt{45}$

Answer: (A)

Radical Expressions with Variables

In mathematics, radicals can involve both numbers and variables, and the principles for simplifying them remain consistent regardless of their composition. When dealing with square roots or cube roots, the goal is to determine what number or variable, when multiplied by itself a certain number of times, yields the original expression. For instance, the square root of \(9\) is \(3\) because \(3^2 = 9\), and the cube root of \(8\) is \(2\) since \(2^3 = 8\).

When introducing variables, such as in the expression \(\sqrt{x^2}\), the square root asks for a value that, when squared, results in \(x^2\). This value is simply \(x\), as \(x \cdot x = x^2\). A key concept to remember is that if the exponent of a variable matches the index of the radical, they effectively cancel each other out, leading to a straightforward simplification.

For more complex expressions, such as \(\sqrt[3]{x^6}\), we can apply the exponent rule where we multiply the exponents. Here, we find that \(\sqrt[3]{x^6} = x^{6/3} = x^2\). This method can be extended to other radicals involving variables, allowing us to simplify expressions like \(\sqrt{x^3}\) by breaking it down into manageable parts.

To simplify \(\sqrt{x^3}\), we can express \(x^3\) as \(x^2 \cdot x\). This allows us to separate the radical into two parts: \(\sqrt{x^2} \cdot \sqrt{x}\). Since \(\sqrt{x^2} = x\), the final result is \(x \cdot \sqrt{x}\).

For higher powers, such as \(\sqrt{x^7}\), we can determine how many times the index (which is \(2\) for square roots) fits into the exponent. In this case, \(2\) goes into \(7\) three times, yielding \(x^3\) as the extracted factor, with \(x\) remaining under the radical. Thus, \(\sqrt{x^7} = x^3 \cdot \sqrt{x}\).

When combining numbers and variables, such as in \(\sqrt{8x^5}\), we can treat them separately. The number \(8\) can be simplified as \(\sqrt{4} \cdot \sqrt{2}\), where \(\sqrt{4} = 2\). For \(x^5\), we apply the same principle: \(2\) goes into \(5\) twice, giving us \(x^2\) outside the radical and leaving \(x\) inside. Therefore, \(\sqrt{8x^5} = 2x^2 \cdot \sqrt{2x}\).

In summary, the process of simplifying radicals with variables mirrors that of simplifying numerical radicals. By breaking down the expressions into their components and applying exponent rules, we can effectively simplify complex radical expressions.

Simplify the radical.
$\sqrt{63x^2}$

• A. $63\sqrt{x}$
• B. $3\sqrt{7x}$
• C. $x\sqrt{63}$
• D. $3x\sqrt7$

Answer: (D)

Radical Expressions with Fractions

Understanding how to simplify radicals, especially when dealing with fractions, is essential in algebra. The key principle is that if a term can be expressed as a product, it can be separated into individual radicals. This technique can also be applied to fractions, allowing us to simplify complex expressions more easily.

When simplifying a radical that contains a fraction, we can break it down into the square root of the numerator and the square root of the denominator. For example, consider the fraction \(\frac{49}{64}\). Both 49 and 64 are perfect squares, so we can express this as \(\sqrt{49} / \sqrt{64}\), which simplifies to \(\frac{7}{8}\).

In another example, simplifying \(\frac{\sqrt{32}}{\sqrt{2}}\) can be approached by combining the radicals into one: \(\sqrt{\frac{32}{2}} = \sqrt{16}\), which simplifies to 4, a perfect square.

When variables are involved, the same principles apply. For instance, consider \(\sqrt{\frac{64x^4}{9x^2}}\). Instead of trying to simplify the fraction directly, we can separate it into \(\frac{\sqrt{64x^4}}{\sqrt{9x^2}}\). Here, \(\sqrt{64} = 8\) and \(\sqrt{x^4} = x^2\), while \(\sqrt{9} = 3\) and \(\sqrt{x^2} = x\). This results in \(\frac{8x^2}{3x}\), which can be further simplified to \(\frac{8}{3}x\) by applying the quotient rule for exponents.

In another scenario, simplifying \(\sqrt{\frac{72x^3}{9x}}\) involves first simplifying the fraction to \(\sqrt{8x^2}\). Recognizing that 8 can be expressed as \(\sqrt{4 \cdot 2}\) and \(\sqrt{x^2} = x\), we can rewrite this as \(2x\sqrt{2}\), which is fully simplified.

In summary, the process of simplifying radicals, especially those involving fractions and variables, relies on recognizing perfect squares and applying the properties of exponents. By breaking down complex expressions into manageable parts, we can achieve a clearer and more simplified result.

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, so \(3 + 8\) becomes \(11\).

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’s also important to remember that you cannot merge separate radicals into one. 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 different indexes, but the constant \(4\) can be combined with other constants if present. Thus, the expression simplifies to \(9\sqrt[3]{x} - \sqrt{x} + 4\).

In summary, when working with radical expressions, focus on identifying like radicals based on their radicands and indexes, combine them accordingly, and avoid merging separate radicals incorrectly. This approach will help you simplify radical expressions accurately.

True or False:
$\sqrt{9+16}$ and $\sqrt9+\sqrt{16}$ are equal.

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

Answer: (B)

Adding & Subtracting Unlike Radicals by Simplifying

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 inside 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 a more complex scenario with \(\sqrt{18}\) and \(\sqrt{50}\), both radicals need simplification. \(\sqrt{18}\) simplifies to \(3\sqrt{2}\) and \(\sqrt{50}\) simplifies to \(5\sqrt{2}\). This results in the expression \(3\sqrt{2} + 5\sqrt{2}\), which combines to \(8\sqrt{2}\).

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

Here’s what students ask on this topic:

How do you simplify the square root of 20?

To simplify the square root of 20, you need to break it into a product of factors where one factor is a perfect square. The number 20 can be factored into 4 and 5, where 4 is a perfect square. So, you can write:

$\sqrt{20}=\sqrt{4}*\sqrt{5}=2*\sqrt{5}$

Therefore, the simplified form of the square root of 20 is 2√5.

How do you simplify the cube root of 54x⁴?

To simplify the cube root of 54x⁴, break it into a product of factors where one factor is a perfect cube. The number 54 can be factored into 27 and 2, where 27 is a perfect cube. For the variable part, x⁴ can be split into x³ and x:

$\sqrt[3]{\mathrm{54x^4}}=\sqrt[3]{27}*\sqrt[3]{2}*\sqrt[3]{\mathrm{x^3}}*\sqrt[3]{\mathrm{x}}=3*\sqrt[3]{2}*\mathrm{x}$

Therefore, the simplified form is 3x√[3]{2x}.

How do you simplify the square root of a fraction like 49/64?

To simplify the square root of a fraction like 49/64, you can break it into the square root of the numerator and the square root of the denominator:

$\sqrt{\frac{49}{64}}=\frac{\sqrt{49}}{\sqrt{64}}=\frac{7}{8}$

Therefore, the simplified form of the square root of 49/64 is 7/8.

How do you add and subtract radical expressions?

To add and subtract radical expressions, you need to combine like radicals, which have the same radicand and index. For example, to simplify 2√x + 4√x + 3:

$2\sqrt{x}+4\sqrt{x}+3=\; (2+4)\sqrt{x}+3=6\sqrt{x}+3$

Only like radicals can be combined. For example, 2√x and 4√x can be combined, but 2√x and 3√y cannot.

How do you simplify the square root of x³?

To simplify the square root of x³, break it into a product of factors where one factor is a perfect square. The exponent 3 can be split into 2 and 1:

$\sqrt{x^3}=\sqrt{x^2}*\sqrt{x}=x*\sqrt{x}$

Therefore, the simplified form of the square root of x³ is x√x.