# Multiplying Radicals Worksheet, Rules, and Examples

Get the free Multiplying Radical Expressions Worksheet and other resources for teaching & understanding how to Multiply Radical Expressions

### Key Points about Multiplying Radicals

- Multiplying radical expressions involves multiplying the radicands together while keeping their product under the same radical symbol.
- Multiplying radicals can involve expressions with variables and coefficients.
- To simplify the expression, it is important to find perfect squares and simplify the radical if possible.

**Multiplying Radical Expressions: A Clear Guide**

Multiplying radicals is a fundamental concept in algebra that involves multiplying expressions that contain square roots or other radicals. It is a crucial skill to master, as it is used extensively in higher-level math courses, including calculus. Multiplying radicals can seem intimidating at first, but with practice, it becomes easier to understand and apply.

When multiplying radical expressions, the key is to multiply the radicands together while keeping their product under the same radical symbol. If the radical expressions have numbers located outside, those numbers can be multiplied separately and then combined with the product of the radicands. It is important to simplify the expression as much as possible by finding perfect squares and simplifying the radical if possible.

Multiplying radicals is not limited to expressions with just numbers. It can also involve multiplying expressions with variables and coefficients. Understanding the rules for multiplying radicals is crucial to solving more complex problems. In this article, we will explore the different rules for multiplying radicals, provide examples, and answer some frequently asked questions.

**Common Core Standard:**

**Related Topics:**Adding and Subtracting Radical Expressions, Dividing Radicals, Using the Distance Formula, Using the Midpoint Formula

**Multiplying Radical Expressions**

Multiplying radical expressions involves applying the product rule for radicals. When multiplying two radicals with the same index, we multiply the numbers outside the radicals and multiply the numbers inside the radicals. For example, when multiplying √a and √b, we get √(ab).

To multiply two radical expressions, we use the FOIL method, which stands for First, Outer, Inner, and Last. The FOIL method is used to multiply two binomials, and it can also be applied to multiplying two radical expressions.

When multiplying two binomials a + b and c + d, we get:

(a + b) * (c + d) = ac + ad + bc + bd

When multiplying two radical expressions √a + √b and √c + √d, we get:

(√a + √b) * (√c + √d) = √ac + √ad + √bc + √bd

Multiplying two radical expressions can also involve multiplying polynomials. In this case, we use the distributive property of multiplication to simplify the expression before applying the product rule for radicals.

For example, to multiply (√2 + 3)(√3 + 4), we first simplify the expression using the distributive property of multiplication:

(√2 + 3)(√3 + 4) = √2 * √3 + √2 * 4 + 3 * √3 + 3 * 4

= √6 + 4√2 + 3√3 + 12

Then, we apply the product rule for radicals to simplify the expression further:

√6 + 4√2 + 3√3 + 12 = (√2 * √3) + (4 * √2) + (3 * √3) + 12

= 2√3 + 4√2 + 3√3 + 12

= 5√3 + 4√2 + 12

In summary, multiplying radical expressions involves applying the product rule for radicals and using the FOIL method or the distributive property of multiplication to simplify the expression before applying the product rule.

**Multiplying Radicals with Different Index**

Multiplying radicals with different indices can be a bit more complicated than multiplying radicals with the same index. However, with a bit of practice, it can be easily accomplished.

When multiplying radicals with different indices, the first step is to simplify the radicals as much as possible. This can be done by breaking down each radical into its prime factors and then grouping those factors together. Once the radicals have been simplified, the next step is to multiply the radicands together.

It is important to note that when multiplying radicals with different indices, the resulting product will have an index that is equal to the product of the two original indices. For example, if you are multiplying a square root with an index of 2 by a cube root with an index of 3, the resulting product will have an index of 6.

Here is an example to illustrate how to multiply radicals with different indices:

√2 * 3√5 = √(2*5^3) = √250 = 5√10

In the example above, the first step is to simplify the radicals. The square root of 2 cannot be simplified any further, but the cube root of 5 can be simplified to 5^(1/3). Then, the radicands are multiplied together to get 2 * 5^(1/3). Finally, the radical is simplified to get 5√10.

It is important to note that not all radicals with different indices can be simplified. In some cases, it may be necessary to leave the expression in radical form.

In summary, multiplying radicals with different indices involves simplifying the radicals as much as possible, multiplying the radicands together, and simplifying the resulting radical expression. With practice, this process can become second nature, allowing you to easily multiply radicals with different indices.

**Multiplying Radicals with Variables**

Multiplying radicals with variables is similar to multiplying radicals without variables. The basic rule is to multiply the coefficients and the radicands separately. However, when multiplying radicals with variables, it is important to simplify the resulting expression by combining like terms.

To multiply two radicals with variables, multiply the coefficients and multiply the radicands separately. For example, to multiply √3x and √2x, first multiply the coefficients 3 and 2, which gives 6. Then, multiply the radicands x and x, which gives x². Therefore, the product of √3x and √2x is 6x².

When multiplying radicals with variables, it is important to simplify the expression by combining like terms. For example, to multiply √3x and 2√3x, first multiply the coefficients 3 and 2, which gives 6. Then, multiply the radicands x and x, which gives x². Therefore, the product of √3x and 2√3x is 6x². However, this expression can be simplified by combining the like terms 6x² and 2√3x. The simplified expression is 6x² + 2√3x.

It is also important to simplify radicals when multiplying radicals with variables. For example, to multiply √2x and √8x, first multiply the coefficients 2 and 8, which gives 16. Then, multiply the radicands x and x, which gives x². Therefore, the product of √2x and √8x is 16x². However, this expression can be simplified by simplifying the radical √8. √8 can be simplified as 2√2. Therefore, the product of √2x and √8x is 32x²√2.

In summary, to multiply radicals with variables, multiply the coefficients and the radicands separately, combine like terms, and simplify radicals when possible. By following these simple rules, one can easily multiply radicals with variables and simplify the resulting expression.

**Multiplying Radicals with Coefficients**

Multiplying radical expressions with coefficients is a fundamental concept in algebra that is used in many applications, including solving quadratic equations and simplifying complex expressions. The process involves multiplying the coefficients and the radicands separately and then combining them under the same radical symbol.

To multiply radicals with coefficients, the first step is to simplify the radicands. If the radicands are not perfect squares, then simplify them as much as possible. Next, multiply the coefficients and the radicands separately, and then combine them under the same radical symbol.

For example, consider the expression 3√6 ⋅ 5√2. Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands separately as follows:

3√6 ⋅ 5√2 = 3 ⋅ 5 ⋅ √6 ⋅ √2 (Multiplication is commutative)

= 15 ⋅ √12 (Multiply the coefficients and the radicands)

= 15√4 ⋅ 3 (Simplify)

= 15 ⋅ 2 ⋅ √3 (Simplify)

= 30√3

It is important to note that when multiplying radicals with coefficients, the coefficients are treated as absolute values. This means that negative coefficients should be converted to positive coefficients before multiplying. For instance, -2√3 should be simplified to -1 ⋅ 2√3, which equals -2√3.

In summary, multiplying radical expressions with coefficients involves simplifying the radicands, multiplying the coefficients and the radicands separately, and then combining them under the same radical symbol. It is a fundamental concept in algebra that is used in many applications, and it is important to understand the process to solve complex expressions.

**Multiplying Radicals Examples**

Multiplying radical expressions involves multiplying the coefficients and the radicands separately. Here are some examples to illustrate the process:

**Example 1:** Multiply √6 by √3.

Solution:

√6 × √3 = √(6 × 3) = √18

Since 18 can be factored as 2 × 3 × 3, we can simplify the expression further:

√18 = √(2 × 3 × 3) = 3√2

Therefore, √6 × √3 = 3√2.

**Example 2:** Multiply 2√5 by 3√7.

Solution:

2√5 × 3√7 = 6√(5 × 7) = 6√35

Therefore, 2√5 × 3√7 = 6√35.

**Example 3:** Multiply (4√3)(5√2).

Solution:

(4√3)(5√2) = 20√(3 × 2) = 20√6

Therefore, (4√3)(5√2) = 20√6.

**Example 4:** Multiply (2 + √3)(3 – √3).

Solution:

(2 + √3)(3 – √3) = 6 – 2√3 + 3√3 – √3 × √3

Note that √3 × √3 is equal to 3, so we can simplify the expression further:

(2 + √3)(3 – √3) = 6 + √3 – 3 = 3 + √3

Therefore, (2 + √3)(3 – √3) = 3 + √3.

By following the above examples, one can multiply radical expressions with ease.

**Rules for Multiplying Radicals**

Multiplying radicals is an important skill in algebra that involves multiplying terms that contain square roots, cube roots, or other types of radicals. Below are the basic rules for multiplying radicals:

- When multiplying radicals with the same index, we can use the product rule for radicals. The product rule states that the product of two radicals is equal to the radical of the product of their radicands. For example, √a * √b = √(ab).
- To multiply radicals with different indices, we need to simplify them first by expressing them with the same index. For example, to multiply √2 and ∛3, we can express them as ∛(2^3) and ∛(3^2), respectively. Then we can multiply them using the product rule.
- When multiplying radicals with variables, we can multiply the coefficients and the variables separately. For example, (2√3)(3√5) = 6√15.
- When multiplying radicals with fractions or fractional exponents, we need to use the power rule for exponents. The power rule states that the product of two terms with the same base raised to different exponents is equal to the base raised to the sum of their exponents. For example, (2√3)^2 = 4*3 = 12.
- When multiplying radicals with like terms, we can combine them by adding their coefficients. For example, 2√5 + 3√5 = 5√5.

In summary, multiplying radicals involves multiplying their radicands together while keeping their product under the same radical symbol. It is important to simplify the radicals first before multiplying them, and to combine like terms if necessary. By following these rules, one can successfully multiply radicals with confidence.

**Multiplying and Dividing Radical Expressions**

Multiplying and dividing radical expressions can be a bit tricky, but it’s an essential skill to master in algebra. In this section, we’ll explore the basics of multiplying and dividing radical expressions.

**Multiplying Radical Expressions**

To multiply radical expressions, we use the product rule for radicals. Given real numbers n√A and n√B, n√A ⋅ n√B = n√A ⋅ B. For example, if we want to multiply 3√12 and 3√6, we can apply the product rule for radicals as follows:

3√12 ⋅ 3√6 = 3√12 ⋅ 6 (multiply the radicands) = 3√72 (simplify) = 3√23 ⋅ 32 (factor) = 23√9 (simplify)

**Dividing Radical Expressions**

To divide radical expressions, we use the quotient rule for radicals. Given real numbers n√A and n√B, n√A / n√B = n√A / B. However, the quotient must be rationalized, which means we need to eliminate any radicals in the denominator. To do this, we can multiply both the numerator and denominator by the conjugate of the denominator. For example, if we want to divide 4√3 by √2, we can rationalize the denominator as follows:

4√3 / √2 = 4√3 / √2 * √2 / √2 (multiply by the conjugate of the denominator) = 4√6 / 2 (simplify) = 2√6 (simplify)

**Multiplying and Dividing Radical Expressions with Variables**

Multiplying and dividing radical expressions with variables is similar to doing so with constants. We just need to make sure that the variables have the same exponent and that we simplify the expression as much as possible. For example, if we want to multiply √x and √x^2y, we can simplify the expression as follows:

√x ⋅ √x^2y = √x * x√y (product rule for radicals) = x√xy (simplify)

Similarly, if we want to divide √x^3 by √xy, we can simplify the expression as follows:

√x^3 / √xy = x√x / √xy (quotient rule for radicals) = x√x / y√x (rationalize the denominator) = x / y (simplify)

In conclusion, multiplying and dividing radical expressions require knowledge of the product and quotient rules for radicals. When dealing with variables, we need to make sure that the variables have the same exponent and simplify the expression as much as possible.

**Rationalizing the Denominator**

When multiplying radical expressions, it is often necessary to simplify the resulting expression by rationalizing the denominator. This involves removing any radicals from the denominator of a fraction by multiplying both the numerator and denominator by a suitable expression that will eliminate the radical.

Rationalizing the denominator is particularly useful when dealing with rational expressions, which are expressions that involve a ratio of polynomials. In these cases, it is important to simplify the expression as much as possible to avoid potential errors when performing further operations.

One common method of rationalizing the denominator is to use conjugates. A conjugate is a binomial expression of the form a + b√c, where a, b, and c are rational numbers. The conjugate of a + b√c is a – b√c, and vice versa. When rationalizing a denominator that contains a radical of the form √c, the conjugate of the denominator is multiplied by both the numerator and denominator of the fraction. This results in the elimination of the radical from the denominator, leaving only rational terms.

For example, consider the expression (2√3)/(√2). To rationalize the denominator, the conjugate of the denominator, √2, is multiplied by both the numerator and denominator of the fraction:

(2√3)/(√2) * (√2)/(√2) = (2√6)/2 = √6

Thus, the simplified expression is √6.

In some cases, it may be necessary to simplify the expression further by factoring out any common factors or simplifying any remaining radicals. It is important to check the final expression to ensure that it is simplified as much as possible.

Overall, rationalizing the denominator is an important technique for simplifying radical expressions and rational expressions. Using conjugates is a common method for achieving this simplification, but it is important to simplify the resulting expression as much as possible to avoid errors in further calculations.

**Multiplying Radicals FAQ**

**How do you multiply radicals with coefficients?**

To multiply radicals with coefficients, you can use the product rule for radicals. First, multiply the coefficients together, and then multiply the radicands together. For example, to multiply 2√3 and 3√2, you would multiply 2 and 3 to get 6, and then multiply √3 and √2 to get √6. So the product of 2√3 and 3√2 is 6√6.

**What are the rules for multiplying radicals?**

The rules for multiplying radicals include the product rule for radicals and the quotient rule for radicals. The product rule states that the product of two radicals is equal to the radical of their product. The quotient rule states that the quotient of two radicals is equal to the radical of their quotient.

**Can you provide an example of multiplying radicals with the same radicand?**

Sure! When multiplying radicals with the same radicand, you can use the product rule for radicals. For example, to multiply √3 and √3, you would multiply the radicands together to get 3. So the product of √3 and √3 is simply 3.

**What is the process for multiplying radicals?**

The process for multiplying radicals involves using the product rule for radicals. To multiply two radicals, you multiply their coefficients together and then multiply their radicands together. If the radicands are the same, you can simply multiply the coefficients and leave the radicand the same.

**Where can I find practice problems for multiplying radicals?**

You can find practice problems for multiplying radicals in algebra textbooks, online math resources, and math workbooks. Many of these resources offer step-by-step solutions to help you understand the process of multiplying radicals.

**What happens when you multiply two radicals with different radicands?**

When you multiply two radicals with different radicands, you cannot simplify the product any further. For example, the product of √2 and √3 cannot be simplified any further, so the answer is simply √6.

**What happens when you multiply two radicals?**

When you multiply two radicals, you use the product rule for radicals to simplify the product. If the radicands are the same, you can simply multiply the coefficients and leave the radicand the same. If the radicands are different, you multiply both the coefficients and the radicands together.

**What are the rules for root multiplication?**

The rules for root multiplication include the product rule for roots and the quotient rule for roots. The product rule states that the product of two roots is equal to the root of their product. The quotient rule states that the quotient of two roots is equal to the root of their quotient.

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