Simplify.$\sqrt{8} \cdot \sqrt{6}$

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Understanding the Problem

When dealing with the multiplication of square roots, it's essential to understand the properties of radicals. The multiplication of square roots can be simplified by multiplying the numbers inside the square roots and then simplifying the resulting expression. In this case, we are given the expression $\sqrt{8} \cdot \sqrt{6}$, and we need to simplify it.

Properties of Radicals

Radicals are a way of expressing the square root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. The square root of a number can be expressed as $\sqrt{x}$, where $x$ is the number inside the radical.

When multiplying radicals, we can use the property that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. This means that we can multiply the numbers inside the radicals and then take the square root of the product.

Simplifying the Expression

To simplify the expression $\sqrt{8} \cdot \sqrt{6}$, we can start by multiplying the numbers inside the radicals. We have $\sqrt{8} \cdot \sqrt{6} = \sqrt{8 \cdot 6}$. Now, we can simplify the product inside the radical.

Breaking Down the Numbers

To simplify the product inside the radical, we can break down the numbers into their prime factors. We have $8 = 2^3$ and $6 = 2 \cdot 3$. Now, we can multiply the prime factors together.

Multiplying the Prime Factors

We have $8 \cdot 6 = 2^3 \cdot 2 \cdot 3$. Now, we can combine the prime factors by adding their exponents. We have $2^3 \cdot 2 \cdot 3 = 2^{3+1} \cdot 3 = 2^4 \cdot 3$.

Simplifying the Radical

Now that we have the product inside the radical, we can simplify the radical by taking the square root of the product. We have $\sqrt{8 \cdot 6} = \sqrt{2^4 \cdot 3}$. Since the square root of $2^4$ is 4, we can simplify the radical as follows:

Final Simplification

We have $\sqrt{2^4 \cdot 3} = 4\sqrt{3}$. Therefore, the simplified expression is $4\sqrt{3}$.

Conclusion

In this article, we simplified the expression $\sqrt{8} \cdot \sqrt{6}$ by multiplying the numbers inside the radicals and then simplifying the resulting expression. We used the property that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ and broke down the numbers into their prime factors to simplify the product inside the radical. The final simplified expression is $4\sqrt{3}$.

Additional Examples

Here are a few additional examples of simplifying the multiplication of square roots:

• $\sqrt{9} \cdot \sqrt{16} = \sqrt{9 \cdot 16} = \sqrt{144} = 12$
• $\sqrt{25} \cdot \sqrt{36} = \sqrt{25 \cdot 36} = \sqrt{900} = 30$
• $\sqrt{49} \cdot \sqrt{64} = \sqrt{49 \cdot 64} = \sqrt{3136} = 56$

Final Thoughts

Simplifying the multiplication of square roots can be a challenging task, but by using the properties of radicals and breaking down the numbers into their prime factors, we can simplify the expression and find the final answer.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about simplifying the multiplication of square roots.

Q: What is the property of radicals that allows us to multiply the numbers inside the radicals?

A: The property of radicals that allows us to multiply the numbers inside the radicals is $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. This means that we can multiply the numbers inside the radicals and then take the square root of the product.

Q: How do we simplify the product inside the radical?

A: To simplify the product inside the radical, we can break down the numbers into their prime factors. We can then combine the prime factors by adding their exponents.

Q: What is the difference between simplifying a radical and simplifying the multiplication of square roots?

A: Simplifying a radical involves finding the square root of a number, while simplifying the multiplication of square roots involves multiplying two or more square roots together. When simplifying the multiplication of square roots, we can use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify the expression.

Q: Can we simplify the multiplication of square roots with different radicands?

A: Yes, we can simplify the multiplication of square roots with different radicands. We can use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify the expression.

Q: How do we handle negative numbers when simplifying the multiplication of square roots?

A: When simplifying the multiplication of square roots, we can handle negative numbers by using the property $\sqrt{-a} = i\sqrt{a}$, where $i$ is the imaginary unit.

Q: Can we simplify the multiplication of square roots with variables?

A: Yes, we can simplify the multiplication of square roots with variables. We can use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify the expression.

Q: What is the final simplified expression for $\sqrt{8} \cdot \sqrt{6}$?

A: The final simplified expression for $\sqrt{8} \cdot \sqrt{6}$ is $4\sqrt{3}$.

Q: Can you provide more examples of simplifying the multiplication of square roots?

A: Yes, here are a few more examples:

• $\sqrt{9} \cdot \sqrt{16} = \sqrt{9 \cdot 16} = \sqrt{144} = 12$
• $\sqrt{25} \cdot \sqrt{36} = \sqrt{25 \cdot 36} = \sqrt{900} = 30$
• $\sqrt{49} \cdot \sqrt{64} = \sqrt{49 \cdot 64} = \sqrt{3136} = 56$

Q: What are some common mistakes to avoid when simplifying the multiplication of square roots?

A: Some common mistakes to avoid when simplifying the multiplication of square roots include:

• Not using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to simplify the expression
• Not breaking down the numbers into their prime factors
• Not combining the prime factors by adding their exponents
• Not handling negative numbers correctly

Q: How can I practice simplifying the multiplication of square roots?

A: You can practice simplifying the multiplication of square roots by working through examples and exercises. You can also use online resources and practice problems to help you improve your skills.

Q: What are some real-world applications of simplifying the multiplication of square roots?

A: Simplifying the multiplication of square roots has many real-world applications, including:

• Calculating distances and lengths in geometry and trigonometry
• Solving equations and inequalities in algebra and calculus
• Working with complex numbers and polynomials in mathematics and engineering
• Modeling real-world phenomena in physics and engineering

Q: Can you provide more information about the history of simplifying the multiplication of square roots?

A: The history of simplifying the multiplication of square roots dates back to ancient civilizations, where mathematicians such as Euclid and Diophantus developed methods for simplifying radicals and multiplying square roots. Over time, mathematicians such as Pierre de Fermat and Leonhard Euler developed more advanced methods for simplifying the multiplication of square roots, which are still used today.