What Is The Following Sum In Simplest Form?A. 8 + 3 2 + 32 \sqrt{8} + 3\sqrt{2} + \sqrt{32} 8 ​ + 3 2 ​ + 32 ​ B. 3 8 + 3 2 3\sqrt{8} + 3\sqrt{2} 3 8 ​ + 3 2 ​ C. 5 42 5\sqrt{42} 5 42 ​ D. 9 2 9\sqrt{2} 9 2 ​ E. 5 2 + 32 5\sqrt{2} + \sqrt{32} 5 2 ​ + 32 ​

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A. $\sqrt{8} + 3\sqrt{2} + \sqrt{32}$

To simplify the given expression, we need to first simplify each radical term individually. We can start by simplifying $\sqrt{8}$ and $\sqrt{32}$.

$\sqrt{8}$ can be simplified as $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$.

Similarly, $\sqrt{32}$ can be simplified as $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$.

Now, we can substitute these simplified values back into the original expression:

$\sqrt{8} + 3\sqrt{2} + \sqrt{32} = 2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2}$

Combining like terms, we get:

$2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} = 9\sqrt{2}$

Therefore, the simplified form of the given expression is $9\sqrt{2}$.

B. $3\sqrt{8} + 3\sqrt{2}$

To simplify the given expression, we need to first simplify each radical term individually. We can start by simplifying $\sqrt{8}$.

$\sqrt{8}$ can be simplified as $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$.

Now, we can substitute this simplified value back into the original expression:

$3\sqrt{8} + 3\sqrt{2} = 3(2\sqrt{2}) + 3\sqrt{2}$

Using the distributive property, we get:

$3(2\sqrt{2}) + 3\sqrt{2} = 6\sqrt{2} + 3\sqrt{2}$

Combining like terms, we get:

$6\sqrt{2} + 3\sqrt{2} = 9\sqrt{2}$

Therefore, the simplified form of the given expression is $9\sqrt{2}$.

C. $5\sqrt{42}$

To simplify the given expression, we need to first simplify the radical term $\sqrt{42}$.

$\sqrt{42}$ can be simplified as $\sqrt{42} = \sqrt{6 \times 7} = \sqrt{6} \times \sqrt{7}$.

However, $\sqrt{6}$ and $\sqrt{7}$ cannot be simplified further, so we cannot simplify the expression $5\sqrt{42}$ any further.

Therefore, the simplified form of the given expression is $5\sqrt{42}$.

D. $9\sqrt{2}$

The given expression is already in its simplest form, so we do not need to simplify it any further.

Therefore, the simplified form of the given expression is $9\sqrt{2}$.

E. $5\sqrt{2} + \sqrt{32}$

To simplify the given expression, we need to first simplify each radical term individually. We can start by simplifying $\sqrt{32}$.

$\sqrt{32}$ can be simplified as $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$.

Now, we can substitute this simplified value back into the original expression:

$5\sqrt{2} + \sqrt{32} = 5\sqrt{2} + 4\sqrt{2}$

Combining like terms, we get:

$5\sqrt{2} + 4\sqrt{2} = 9\sqrt{2}$

Therefore, the simplified form of the given expression is $9\sqrt{2}$.

Conclusion

In conclusion, the simplified forms of the given expressions are:

• A. $\sqrt{8} + 3\sqrt{2} + \sqrt{32} = 9\sqrt{2}$
• B. $3\sqrt{8} + 3\sqrt{2} = 9\sqrt{2}$
• C. $5\sqrt{42} = 5\sqrt{42}$
• D. $9\sqrt{2} = 9\sqrt{2}$
• E. $5\sqrt{2} + \sqrt{32} = 9\sqrt{2}$

Therefore, the correct answer is A, B, D, and E, which all simplify to $9\sqrt{2}$.

Frequently Asked Questions

Q: What is the process for simplifying radical expressions?

A: To simplify radical expressions, we need to first simplify each radical term individually. We can start by simplifying the radical term using the properties of radicals, such as the product rule and the quotient rule.

Q: How do I simplify a radical term?

A: To simplify a radical term, we need to find the largest perfect square that divides the number inside the radical. We can then rewrite the radical term as the product of the perfect square and the remaining number.

Q: What is the difference between a simplified radical expression and a non-simplified radical expression?

A: A simplified radical expression is one in which the radical term has been simplified using the properties of radicals, whereas a non-simplified radical expression is one in which the radical term has not been simplified.

Q: Can I simplify a radical expression that contains multiple radical terms?

A: Yes, you can simplify a radical expression that contains multiple radical terms by simplifying each radical term individually and then combining the simplified terms.

Q: How do I know when a radical expression is in its simplest form?

A: A radical expression is in its simplest form when the radical term cannot be simplified further using the properties of radicals.

Q: Can I simplify a radical expression that contains a variable?

A: Yes, you can simplify a radical expression that contains a variable by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the importance of simplifying radical expressions?

A: Simplifying radical expressions is important because it allows us to work with simpler expressions and to perform calculations more easily.

Q: Can I use a calculator to simplify radical expressions?

A: Yes, you can use a calculator to simplify radical expressions, but it is also important to understand the underlying math and to be able to simplify radical expressions by hand.

Q: How do I know when to use the product rule and when to use the quotient rule to simplify radical expressions?

A: You should use the product rule when simplifying a radical expression that contains multiple radical terms, and you should use the quotient rule when simplifying a radical expression that contains a fraction.

Q: Can I simplify a radical expression that contains a negative number?

A: Yes, you can simplify a radical expression that contains a negative number by simplifying the radical term using the properties of radicals and then combining the simplified terms.

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

A: A rational number is a number that can be expressed as the ratio of two integers, whereas an irrational number is a number that cannot be expressed as the ratio of two integers.

Q: Can I simplify a radical expression that contains a rational number?

A: Yes, you can simplify a radical expression that contains a rational number by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: How do I know when to use the distributive property to simplify radical expressions?

A: You should use the distributive property when simplifying a radical expression that contains multiple terms.

Q: Can I simplify a radical expression that contains a variable and a constant?

A: Yes, you can simplify a radical expression that contains a variable and a constant by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the importance of understanding the properties of radicals?

A: Understanding the properties of radicals is important because it allows us to simplify radical expressions and to perform calculations more easily.

Q: Can I use a calculator to check my work when simplifying radical expressions?

A: Yes, you can use a calculator to check your work when simplifying radical expressions, but it is also important to understand the underlying math and to be able to simplify radical expressions by hand.

Q: How do I know when to use the order of operations to simplify radical expressions?

A: You should use the order of operations when simplifying a radical expression that contains multiple terms and operations.

Q: Can I simplify a radical expression that contains a negative variable?

A: Yes, you can simplify a radical expression that contains a negative variable by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the difference between a simplified radical expression and a non-simplified radical expression in terms of their decimal equivalents?

A: A simplified radical expression will have a decimal equivalent that is a perfect square, whereas a non-simplified radical expression will have a decimal equivalent that is not a perfect square.

Q: Can I simplify a radical expression that contains a variable and a radical term?

A: Yes, you can simplify a radical expression that contains a variable and a radical term by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: How do I know when to use the conjugate to simplify radical expressions?

A: You should use the conjugate when simplifying a radical expression that contains a binomial.

Q: Can I simplify a radical expression that contains a variable and a constant and a radical term?

A: Yes, you can simplify a radical expression that contains a variable and a constant and a radical term by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the importance of understanding the properties of radicals in terms of their applications?

A: Understanding the properties of radicals is important because it allows us to simplify radical expressions and to perform calculations more easily, which has many applications in mathematics and science.

Q: Can I use a calculator to check my work when simplifying radical expressions that contain variables?

A: Yes, you can use a calculator to check your work when simplifying radical expressions that contain variables, but it is also important to understand the underlying math and to be able to simplify radical expressions by hand.

Q: How do I know when to use the product rule and when to use the quotient rule to simplify radical expressions that contain variables?

A: You should use the product rule when simplifying a radical expression that contains multiple radical terms and variables, and you should use the quotient rule when simplifying a radical expression that contains a fraction and variables.

Q: Can I simplify a radical expression that contains a variable and a radical term and a constant?

A: Yes, you can simplify a radical expression that contains a variable and a radical term and a constant by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the difference between a simplified radical expression and a non-simplified radical expression in terms of their algebraic equivalents?

A: A simplified radical expression will have an algebraic equivalent that is a perfect square, whereas a non-simplified radical expression will have an algebraic equivalent that is not a perfect square.

Q: Can I simplify a radical expression that contains a variable and a radical term and a fraction?

A: Yes, you can simplify a radical expression that contains a variable and a radical term and a fraction by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: How do I know when to use the distributive property to simplify radical expressions that contain variables?

A: You should use the distributive property when simplifying a radical expression that contains multiple terms and variables.

Q: Can I simplify a radical expression that contains a variable and a radical term and a negative number?

A: Yes, you can simplify a radical expression that contains a variable and a radical term and a negative number by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the importance of understanding the properties of radicals in terms of their applications in mathematics and science?

A: Understanding the properties of radicals is important because it allows us to simplify radical expressions and to perform calculations more easily, which has many applications in mathematics and science.

Q: Can I use a calculator to check my work when simplifying radical expressions that contain variables and fractions?

A: Yes, you can use a calculator to check your work when simplifying radical expressions that contain variables and fractions, but it is also important to understand the underlying math and to be able to simplify radical expressions by hand.

Q: How do I know when to use the order of operations to simplify radical expressions that contain variables and fractions?

A: You should use the order of operations when simplifying a radical expression that contains multiple terms and variables and fractions.

Q: Can I simplify a radical expression that contains a variable and a radical term and a negative variable?

A: Yes, you can simplify a radical expression that contains a variable and a radical term and a negative variable by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: What is the difference between a simplified radical expression and a non-simplified radical expression in terms of their decimal equivalents and algebraic equivalents?

A: A simplified radical expression will have a decimal equivalent that is a perfect square and an algebraic equivalent that is a perfect square, whereas a non-simplified radical expression will have a decimal equivalent that is not a perfect square and an algebraic equivalent that is not a perfect square.

Q: Can I simplify a radical expression that contains a variable and a radical term and a fraction and a negative number?

A: Yes, you can simplify a radical expression that contains a variable and a radical term and a fraction and a negative number by simplifying the radical term using the properties of radicals and then combining the simplified terms.

Q: How do I know when to use the conjugate to simplify radical expressions that contain variables and fractions?

A: You should use the conjugate when simplifying a radical expression that contains a binomial and variables and fractions.

Q: Can I simplify a radical expression that contains a variable and a radical term and a fraction and a negative variable?

A: Yes