What Is The Product Of 2 And 4 147 4 \sqrt{147} 4 147 ​ In Simplest Radical Form?

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Understanding the Problem

When dealing with expressions involving radicals, it's essential to simplify them to their simplest form. In this case, we're given the expression $2 \times 4 \sqrt{147}$ and asked to find its product in simplest radical form. To begin, let's break down the given expression and understand its components.

Breaking Down the Expression

The given expression is a product of two terms: $2$ and $4 \sqrt{147}$. We can start by simplifying the second term, which involves a radical. The expression $4 \sqrt{147}$ can be rewritten as $4 \sqrt{3 \times 49}$, since $147 = 3 \times 49$. This allows us to simplify the radical further.

Simplifying the Radical

To simplify the radical, we can use the property of radicals that states $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. Applying this property to the expression $4 \sqrt{3 \times 49}$, we get:

$$4 \sqrt{3 \times 49} = 4 \sqrt{3} \times \sqrt{49}$$

Now, we can simplify the expression $\sqrt{49}$, which is equal to $7$, since $7^2 = 49$. Therefore, the expression becomes:

$$4 \sqrt{3 \times 49} = 4 \sqrt{3} \times 7$$

Simplifying the Expression Further

Now that we have simplified the radical, we can rewrite the original expression as:

$$2 \times 4 \sqrt{147} = 2 \times 4 \sqrt{3} \times 7$$

We can further simplify this expression by combining the constants:

$$2 \times 4 \sqrt{147} = 8 \times 7 \sqrt{3}$$

Final Simplification

The expression $8 \times 7 \sqrt{3}$ can be rewritten as $56 \sqrt{3}$, since $8 \times 7 = 56$. Therefore, the product of $2$ and $4 \sqrt{147}$ in simplest radical form is:

$$56 \sqrt{3}$$

Conclusion

In this article, we have simplified the expression $2 \times 4 \sqrt{147}$ to its simplest radical form. We started by breaking down the expression and simplifying the radical using the properties of radicals. Finally, we combined the constants and simplified the expression to obtain the final result.

Frequently Asked Questions

• What is the product of $2$ and $4 \sqrt{147}$?
• How do you simplify a radical expression?
• What is the simplest radical form of $4 \sqrt{147}$?

Answer Key

• The product of $2$ and $4 \sqrt{147}$ is $56 \sqrt{3}$.
• To simplify a radical expression, you can use the properties of radicals, such as $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.
• The simplest radical form of $4 \sqrt{147}$ is $4 \sqrt{3} \times 7$, which can be rewritten as $28 \sqrt{3}$.

Additional Resources

• For more information on simplifying radical expressions, see the article on "Simplifying Radical Expressions".
• For a list of common radical expressions and their simplified forms, see the article on "Common Radical Expressions".

References

• [1] "Simplifying Radical Expressions" by [Author's Name]
• [2] "Common Radical Expressions" by [Author's Name]

Note: The references provided are fictional and for demonstration purposes only.

Frequently Asked Questions

In this article, we will address some of the most common questions related to simplifying radical expressions. Whether you're a student, a teacher, or simply someone looking to improve your math skills, this article is for you.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root. For example, $\sqrt{16}$ is a radical expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the properties of radicals, such as $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. You can also use the property that $\sqrt{a^2} = a$.

Q: What is the difference between a simplified radical expression and a radical expression in simplest form?

A: A simplified radical expression is one that has been reduced to its simplest form using the properties of radicals. For example, $\sqrt{16}$ is a simplified radical expression because it cannot be reduced further. On the other hand, a radical expression in simplest form is one that has been reduced to its simplest form using the properties of radicals, but may still contain a radical. For example, $\sqrt{3}$ is a radical expression in simplest form because it cannot be reduced further.

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

A: To simplify a radical expression with a coefficient, you can use the property that $\sqrt{a^2} = a$. For example, to simplify the expression $3\sqrt{16}$, you can rewrite it as $3\sqrt{4^2}$, which simplifies to $3 \times 4 = 12$.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. For example, to simplify the expression $\sqrt{x^2}$, you can use the property that $\sqrt{a^2} = a$, which simplifies the expression to $x$.

Q: How do I simplify a radical expression with a negative number?

A: To simplify a radical expression with a negative number, you can use the property that $\sqrt{-a} = i\sqrt{a}$, where $i$ is the imaginary unit.

Q: Can I simplify a radical expression with a fraction?

A: Yes, you can simplify a radical expression with a fraction. For example, to simplify the expression $\sqrt{\frac{1}{4}}$, you can rewrite it as $\frac{\sqrt{1}}{\sqrt{4}}$, which simplifies to $\frac{1}{2}$.

Q: How do I simplify a radical expression with a decimal?

A: To simplify a radical expression with a decimal, you can use the property that $\sqrt{a} = b$ if and only if $a = b^2$. For example, to simplify the expression $\sqrt{0.25}$, you can rewrite it as $\sqrt{\frac{1}{4}}$, which simplifies to $\frac{1}{2}$.

Q: Can I simplify a radical expression with a negative decimal?

A: Yes, you can simplify a radical expression with a negative decimal. For example, to simplify the expression $\sqrt{-0.25}$, you can rewrite it as $i\sqrt{0.25}$, which simplifies to $i\frac{1}{2}$.

Q: How do I simplify a radical expression with a mixed number?

A: To simplify a radical expression with a mixed number, you can use the property that $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive numbers.

Q: Can I simplify a radical expression with a complex number?

A: Yes, you can simplify a radical expression with a complex number. For example, to simplify the expression $\sqrt{3 + 4i}$, you can use the property that $\sqrt{a + bi} = \sqrt{\frac{\sqrt{a^2 + b^2} + a}{2}} + i\sqrt{\frac{\sqrt{a^2 + b^2} - a}{2}}$.

Q: How do I simplify a radical expression with a radical in the denominator?

A: To simplify a radical expression with a radical in the denominator, you can use the property that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.

Q: Can I simplify a radical expression with a radical in the numerator and denominator?

A: Yes, you can simplify a radical expression with a radical in the numerator and denominator. For example, to simplify the expression $\frac{\sqrt{a}}{\sqrt{b}}$, you can use the property that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.

Q: How do I simplify a radical expression with multiple radicals?

A: To simplify a radical expression with multiple radicals, you can use the property that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.

Q: Can I simplify a radical expression with a radical and a power?

A: Yes, you can simplify a radical expression with a radical and a power. For example, to simplify the expression $\sqrt{a^2}$, you can use the property that $\sqrt{a^2} = a$.

Q: How do I simplify a radical expression with a radical and a fraction?

A: To simplify a radical expression with a radical and a fraction, you can use the property that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Q: Can I simplify a radical expression with a radical and a decimal?

A: Yes, you can simplify a radical expression with a radical and a decimal. For example, to simplify the expression $\sqrt{0.25}$, you can rewrite it as $\sqrt{\frac{1}{4}}$, which simplifies to $\frac{1}{2}$.

Q: How do I simplify a radical expression with a radical and a negative number?

A: To simplify a radical expression with a radical and a negative number, you can use the property that $\sqrt{-a} = i\sqrt{a}$, where $i$ is the imaginary unit.

Q: Can I simplify a radical expression with a radical and a complex number?

A: Yes, you can simplify a radical expression with a radical and a complex number. For example, to simplify the expression $\sqrt{3 + 4i}$, you can use the property that $\sqrt{a + bi} = \sqrt{\frac{\sqrt{a^2 + b^2} + a}{2}} + i\sqrt{\frac{\sqrt{a^2 + b^2} - a}{2}}$.

Q: How do I simplify a radical expression with multiple radicals and powers?

A: To simplify a radical expression with multiple radicals and powers, you can use the properties of radicals and powers, such as $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ and $\sqrt{a^2} = a$.

Q: Can I simplify a radical expression with multiple radicals and fractions?

A: Yes, you can simplify a radical expression with multiple radicals and fractions. For example, to simplify the expression $\sqrt{\frac{a}{b}}$, you can use the property that $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Q: How do I simplify a radical expression with multiple radicals and decimals?

A: To simplify a radical expression with multiple radicals and decimals, you can use the properties of radicals and decimals, such as $\sqrt{a} = b$ if and only if $a = b^2$.

Q: Can I simplify a radical expression with multiple radicals and negative numbers?

A: Yes, you can simplify a radical expression with multiple radicals and negative numbers. For example, to simplify the expression $\sqrt{-a}$, you can use the property that $\sqrt{-a} = i\sqrt{a}$, where $i$ is the imaginary unit.

Q: How do I simplify a radical expression with multiple radicals and complex numbers?

A: To simplify a radical expression with multiple radicals and complex numbers, you can use the properties of radicals and complex numbers, such as $\sqrt{a + bi} = \sqrt{\frac{\sqrt{a^2 + b^2} + a}{2}} + i\sqrt{\frac{\sqrt{a^2 + b^2} - a}{2}}$.

Q: Can I simplify a radical expression with multiple radicals and powers, fractions, decimals, negative numbers, and complex numbers?

A: Yes, you can simplify a radical expression with multiple radicals and powers, fractions, decimals, negative numbers, and complex numbers. For example, to simplify the expression $\sqrt{\frac{a}{b}}$, you can use the properties of radicals, powers, fractions, decimals, negative numbers, and complex numbers.

Conclusion

In this article, we have addressed some of the most common questions related to simplifying radical expressions. Whether you're a student, a teacher, or simply someone looking to improve your math skills, this article is for you. We have covered a wide range of topics, from simplifying radical expressions with coefficients and variables to simplifying radical expressions with multiple radicals and complex numbers. We hope that this article has been helpful in answering your questions and providing you with the tools you need to simplify radical expressions.