Properties of Multiplication with Radicals

Oct 13, 2025 by windailys.com 146 views

Understanding the Problem

When dealing with radicals, it's essential to understand the properties of multiplication and how they apply to radical expressions. In this case, we're given two radical expressions, $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$, and we're asked to find their product in simplest radical form.

Properties of Multiplication with Radicals

To find the product of two radical expressions, we can use the property of multiplication that states:

a \sqrt{b} \cdot c \sqrt{d} = ac \sqrt{bd}

This property allows us to multiply the coefficients (the numbers in front of the radicals) and multiply the radicands (the numbers inside the radicals).

Applying the Property to the Given Expressions

Using the property of multiplication with radicals, we can find the product of $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$ as follows:

7 \sqrt{6} \cdot 7 \sqrt{14} = 7 \cdot 7 \sqrt{6 \cdot 14}

Simplifying the Radicand

Now, let's simplify the radicand by multiplying the numbers inside the radical:

6 \cdot 14 = 84

So, the product becomes:

7 \cdot 7 \sqrt{84}

Simplifying the Product

Next, let's simplify the product by multiplying the coefficients:

7 \cdot 7 = 49

So, the product becomes:

49 \sqrt{84}

Simplifying the Radical

Now, let's simplify the radical by finding the largest perfect square that divides the radicand:

84 = 4 \cdot 21

Since 4 is a perfect square, we can rewrite the radical as:

\sqrt{84} = \sqrt{4 \cdot 21} = \sqrt{4} \cdot \sqrt{21} = 2 \sqrt{21}

Final Answer

Therefore, the product of $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$ in simplest radical form is:

49 \cdot 2 \sqrt{21} = 98 \sqrt{21}

Conclusion

In this article, we used the property of multiplication with radicals to find the product of $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$ in simplest radical form. We simplified the radicand, multiplied the coefficients, and simplified the radical to arrive at the final answer.

Frequently Asked Questions

What is the product of $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$?
How do you simplify a radical expression?
What is the property of multiplication with radicals?

Answering the FAQs

The product of $$7 \sqrt{6}$$ and $$7 \sqrt{14}$$ is $$98 \sqrt{21}$$.
To simplify a radical expression, you need to find the largest perfect square that divides the radicand and rewrite the radical as the product of that perfect square and the remaining factor.
The property of multiplication with radicals states that $$a \sqrt{b} \cdot c \sqrt{d} = ac \sqrt{bd}$$.

Additional Resources

For more information on radicals and their properties, check out the following resources:

Khan Academy: Radicals
Mathway: Radicals
Wolfram MathWorld: Radicals

Final Thoughts

In conclusion, finding the product of radical expressions requires a good understanding of the properties of multiplication with radicals. By simplifying the radicand, multiplying the coefficients, and simplifying the radical, we can arrive at the final answer.

Understanding Radicals and Their Properties

Radicals are an essential part of mathematics, and understanding their properties is crucial for solving problems involving radical expressions. In this article, we'll answer some frequently asked questions about radicals and their properties.

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical, which is a symbol that indicates the root of a number. For example, $$\sqrt{4}$$ is a radical expression because it represents the square root of 4.

Q: What is the difference between a radical and an exponent?

A: A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. An exponent is used to represent repeated multiplication of a number by itself, while a radical is used to represent the root of a number. For example, $$2^3$$ is an exponent because it represents 2 multiplied by itself 3 times, while $$\sqrt{4}$$ is a radical because it represents the square root of 4.

Q: What is the property of multiplication with radicals?

A: The property of multiplication with radicals states that $$a \sqrt{b} \cdot c \sqrt{d} = ac \sqrt{bd}$$. This property allows us to multiply the coefficients (the numbers in front of the radicals) and multiply the radicands (the numbers inside the radicals).

Q: How do you simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the radicand and rewrite the radical as the product of that perfect square and the remaining factor. For example, $$\sqrt{12}$$ can be simplified as $$\sqrt{4 \cdot 3} = 2 \sqrt{3}$$.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, while an irrational number is a number that cannot be expressed as the ratio of two integers. For example, $$\frac{1}{2}$$ is a rational number because it can be expressed as the ratio of 1 and 2, while $$\sqrt{2}$$ is an irrational number because it cannot be expressed as the ratio of two integers.

Q: How do you add and subtract radical expressions?

A: To add and subtract radical expressions, you need to have the same radicand and the same coefficient. For example, $$\sqrt{4} + \sqrt{4} = 2 \sqrt{4}$$, but $$\sqrt{4} + \sqrt{9}$$ cannot be simplified because the radicands are different.

Q: What is the property of addition with radicals?

A: The property of addition with radicals states that $$a \sqrt{b} + c \sqrt{d} = a \sqrt{b} + c \sqrt{d}$$ if and only if $$a = c$$ and $$b = d$$. This property allows us to add radical expressions only if the radicands and coefficients are the same.

Q: How do you multiply radical expressions with different radicands?

A: To multiply radical expressions with different radicands, you need to multiply the coefficients and multiply the radicands. For example, $$\sqrt{4} \cdot \sqrt{9} = \sqrt{4 \cdot 9} = \sqrt{36} = 6$$.

Q: What is the property of division with radicals?

A: The property of division with radicals states that $$\frac{a \sqrt{b}}{c \sqrt{d}} = \frac{a}{c} \sqrt{\frac{b}{d}}$$. This property allows us to divide radical expressions by multiplying the coefficients and dividing the radicands.

Q: How do you simplify a radical expression with a coefficient?

A: To simplify a radical expression with a coefficient, you need to find the largest perfect square that divides the radicand and rewrite the radical as the product of that perfect square and the remaining factor. For example, $$3 \sqrt{12}$$ can be simplified as $$3 \cdot 2 \sqrt{3} = 6 \sqrt{3}$$.

Q: What is the difference between a radical and a rational expression?

A: A radical expression is an expression that contains a radical, while a rational expression is an expression that contains a fraction. For example, $$\sqrt{4}$$ is a radical expression because it contains a radical, while $$\frac{1}{2}$$ is a rational expression because it contains a fraction.

Q: How do you add and subtract rational expressions?

A: To add and subtract rational expressions, you need to have the same denominator and the same coefficient. For example, $$\frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$$, but $$\frac{1}{2} + \frac{1}{3}$$ cannot be simplified because the denominators are different.

Q: What is the property of addition with rational expressions?

A: The property of addition with rational expressions states that $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$. This property allows us to add rational expressions by finding a common denominator and adding the numerators.

Q: How do you multiply rational expressions?

A: To multiply rational expressions, you need to multiply the numerators and multiply the denominators. For example, $$\frac{1}{2} \cdot \frac{1}{3} = \frac{1 \cdot 1}{2 \cdot 3} = \frac{1}{6}$$.

Q: What is the property of multiplication with rational expressions?

A: The property of multiplication with rational expressions states that $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$. This property allows us to multiply rational expressions by multiplying the numerators and denominators.

Q: How do you divide rational expressions?

A: To divide rational expressions, you need to multiply the first expression by the reciprocal of the second expression. For example, $$\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \cdot \frac{3}{1} = \frac{3}{2}$$.

Q: What is the property of division with rational expressions?

A: The property of division with rational expressions states that $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$. This property allows us to divide rational expressions by multiplying the first expression by the reciprocal of the second expression.

Conclusion

In this article, we've answered some frequently asked questions about radicals and their properties. We've covered topics such as simplifying radical expressions, adding and subtracting radical expressions, multiplying and dividing radical expressions, and more. We hope this article has been helpful in understanding the properties of radicals and how to apply them to solve problems involving radical expressions.