Multiply. Assume $b$ And $c$ Are Greater Than Or Equal To Zero. 7 B 5 C 2 ⋅ 35 C 4 \sqrt{7 B^5 C^2} \cdot \sqrt{35 C^4} 7 B 5 C 2 ​ ⋅ 35 C 4 ​ □ \square □

Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of radicals and how to simplify them. In this case, we're given the expression $\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4}$ and asked to multiply it. To simplify this expression, we need to apply the properties of radicals and exponents.

Properties of Radicals

Before we dive into simplifying the expression, let's review the properties of radicals. The product of two square roots is equal to the square root of the product of the numbers inside the square roots. Mathematically, this can be represented as:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

This property will be useful in simplifying the given expression.

Simplifying the Expression

Now that we've reviewed the properties of radicals, let's simplify the expression $\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4}$. To do this, we'll apply the property of radicals mentioned earlier.

$$\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4} = \sqrt{7 b^5 c^2 \cdot 35 c^4}$$

Next, we'll simplify the expression inside the square root by multiplying the numbers and combining the exponents.

$$\sqrt{7 b^5 c^2 \cdot 35 c^4} = \sqrt{245 b^5 c^6}$$

Now, we can simplify the expression further by factoring out perfect squares.

$$\sqrt{245 b^5 c^6} = \sqrt{5^2 \cdot 7 \cdot b^5 \cdot c^6}$$

Using the property of radicals, we can rewrite the expression as:

$$\sqrt{5^2 \cdot 7 \cdot b^5 \cdot c^6} = 5 b^2 c^3 \sqrt{7 b c}$$

Conclusion

In this article, we've simplified the radical expression $\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4}$ using the properties of radicals and exponents. By applying the property of radicals, we were able to simplify the expression inside the square root and factor out perfect squares. The final simplified expression is $5 b^2 c^3 \sqrt{7 b c}$.

Key Takeaways

• The product of two square roots is equal to the square root of the product of the numbers inside the square roots.
• To simplify a radical expression, we need to apply the properties of radicals and exponents.
• Factoring out perfect squares can help simplify radical expressions.

Real-World Applications

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It's used in various real-world applications, such as:

• Calculating distances and lengths in geometry
• Solving equations in algebra
• Working with complex numbers in mathematics and engineering

Practice Problems

To practice simplifying radical expressions, try the following problems:

• Simplify the expression $\sqrt{2 x^3 y^2} \cdot \sqrt{18 x^4 y^3}$
• Simplify the expression $\sqrt{5 a^2 b^3} \cdot \sqrt{20 a^4 b^5}$
• Simplify the expression $\sqrt{3 c^2 d^4} \cdot \sqrt{12 c^3 d^6}$

Glossary

• Radical: A symbol used to represent the square root of a number.
• Exponent: A number that represents the power to which a base number is raised.
• Perfect square: A number that can be expressed as the square of an integer.

References

• [1] "Algebra" by Michael Artin
• [2] "Geometry" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Multiply: Q&A ================

Frequently Asked Questions

In this article, we'll answer some frequently asked questions about multiplying radical expressions.

Q: What is the product of two square roots?

A: The product of two square roots is equal to the square root of the product of the numbers inside the square roots. Mathematically, this can be represented as:

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the properties of radicals and exponents. Here are the steps:

1. Multiply the numbers inside the square roots.
2. Combine the exponents.
3. Factor out perfect squares.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2.

Q: How do I simplify the expression $\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4}$?

A: To simplify this expression, you need to apply the properties of radicals and exponents. Here are the steps:

1. Multiply the numbers inside the square roots: $\sqrt{7 b^5 c^2} \cdot \sqrt{35 c^4} = \sqrt{7 b^5 c^2 \cdot 35 c^4}$
2. Combine the exponents: $\sqrt{7 b^5 c^2 \cdot 35 c^4} = \sqrt{245 b^5 c^6}$
3. Factor out perfect squares: $\sqrt{245 b^5 c^6} = \sqrt{5^2 \cdot 7 \cdot b^5 \cdot c^6} = 5 b^2 c^3 \sqrt{7 b c}$

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. It's used in various real-world applications, such as:

• Calculating distances and lengths in geometry
• Solving equations in algebra
• Working with complex numbers in mathematics and engineering

Q: How do I practice simplifying radical expressions?

A: To practice simplifying radical expressions, try the following problems:

• Simplify the expression $\sqrt{2 x^3 y^2} \cdot \sqrt{18 x^4 y^3}$
• Simplify the expression $\sqrt{5 a^2 b^3} \cdot \sqrt{20 a^4 b^5}$
• Simplify the expression $\sqrt{3 c^2 d^4} \cdot \sqrt{12 c^3 d^6}$

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Here are some common mistakes to avoid when simplifying radical expressions:

• Not combining the exponents correctly
• Not factoring out perfect squares
• Not using the properties of radicals correctly

Q: How do I check my work when simplifying radical expressions?

A: To check your work when simplifying radical expressions, try the following:

• Plug in values for the variables to see if the expression simplifies correctly
• Use a calculator to check the expression
• Check the expression against a known solution

Conclusion

In this article, we've answered some frequently asked questions about multiplying radical expressions. We've covered topics such as the product of two square roots, simplifying radical expressions, and real-world applications. We've also provided some practice problems and common mistakes to avoid.