Which Expression Is Equivalent To $\sqrt{27}$?A. $3^{\frac{2}{3}}$B. $3^{\frac{3}{2}}$C. $3^{\frac{1}{5}}$D. $3^{\frac{3}{2}}$

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\sqrt{27}$ and determining which of the given options is equivalent to it.

What is a Radical Expression?

A radical expression is a mathematical expression that contains a root or a radical sign. The most common type of radical expression is the square root, denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ is a radical expression because it contains a square root sign.

Simplifying Radical Expressions

Simplifying radical expressions involves expressing them in a simpler form. This can be done by using various mathematical techniques, such as factoring, multiplying, and dividing. In the case of the expression $\sqrt{27}$, we can simplify it by factoring the number 27.

Factoring 27

The number 27 can be factored as follows:

27 = 3 × 3 × 3

This means that 27 can be expressed as the product of three factors of 3.

Simplifying $\sqrt{27}$

Using the factored form of 27, we can simplify the expression $\sqrt{27}$ as follows:

$\sqrt{27}$ = $\sqrt{3 × 3 × 3}$

Using the property of radicals that allows us to separate the product of two or more numbers under the radical sign, we can rewrite the expression as:

$\sqrt{27}$ = $\sqrt{3} × \sqrt{3} × \sqrt{3}$

This can be further simplified by combining the three square roots:

$\sqrt{27}$ = $3\sqrt{3}$

Which Expression is Equivalent to $\sqrt{27}$?

Now that we have simplified the expression $\sqrt{27}$ to $3\sqrt{3}$, we can compare it to the given options to determine which one is equivalent to it.

Option A: $3^{\frac{2}{3}}$

This option can be rewritten as:

$3^{\frac{2}{3}}$ = $\sqrt[3]{3^2}$

Using the property of radicals that allows us to rewrite a radical expression as a fractional exponent, we can rewrite this expression as:

$3^{\frac{2}{3}}$ = $\frac{3^2}{3}$

This simplifies to:

$3^{\frac{2}{3}}$ = $\frac{9}{3}$

Which equals:

$3^{\frac{2}{3}}$ = 3

This is not equivalent to $3\sqrt{3}$.

Option B: $3^{\frac{3}{2}}$

This option can be rewritten as:

$3^{\frac{3}{2}}$ = $\sqrt{3^3}$

Using the property of radicals that allows us to rewrite a radical expression as a fractional exponent, we can rewrite this expression as:

$3^{\frac{3}{2}}$ = $\frac{3^3}{\sqrt{3}}$

This simplifies to:

$3^{\frac{3}{2}}$ = $\frac{27}{\sqrt{3}}$

Which equals:

$3^{\frac{3}{2}}$ = $3\sqrt{3}$

This is equivalent to $3\sqrt{3}$.

Conclusion

In this article, we simplified the expression $\sqrt{27}$ to $3\sqrt{3}$ and compared it to the given options to determine which one is equivalent to it. We found that option B, $3^{\frac{3}{2}}$, is equivalent to $3\sqrt{3}$.

Final Answer

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Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign. The most common type of radical expression is the square root, denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ is a radical expression because it contains a square root sign.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use various mathematical techniques, such as factoring, multiplying, and dividing. You can also use the property of radicals that allows you to separate the product of two or more numbers under the radical sign.

Q: What is the difference between a square root and a cube root?

A: A square root is a radical expression that contains a root of 2, denoted by the symbol $\sqrt{}$. For example, $\sqrt{16}$ is a square root because it contains a root of 2. A cube root, on the other hand, is a radical expression that contains a root of 3, denoted by the symbol $\sqrt[3]{}$. For example, $\sqrt[3]{27}$ is a cube root because it contains a root of 3.

Q: How do I simplify a cube root expression?

A: To simplify a cube root expression, you can use the property of radicals that allows you to separate the product of two or more numbers under the radical sign. You can also use the fact that $\sqrt[3]{a^3} = a$.

Q: What is the relationship between radical expressions and fractional exponents?

A: Radical expressions and fractional exponents are related in that they can be used to represent the same mathematical concept. For example, the expression $\sqrt{a}$ can be rewritten as $a^{\frac{1}{2}}$, and the expression $\sqrt[3]{a}$ can be rewritten as $a^{\frac{1}{3}}$.

Q: How do I convert a radical expression to a fractional exponent?

A: To convert a radical expression to a fractional exponent, you can use the property of radicals that allows you to rewrite a radical expression as a fractional exponent. For example, the expression $\sqrt{a}$ can be rewritten as $a^{\frac{1}{2}}$, and the expression $\sqrt[3]{a}$ can be rewritten as $a^{\frac{1}{3}}$.

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

A: A rational exponent is a fractional exponent that represents a power of a number. For example, $a^{\frac{1}{2}}$ is a rational exponent because it represents a power of the number $a$. A radical expression, on the other hand, is a mathematical expression that contains a root or a radical sign. For example, $\sqrt{a}$ is a radical expression because it contains a square root sign.

Q: How do I simplify a rational exponent expression?

A: To simplify a rational exponent expression, you can use the property of exponents that allows you to multiply and divide exponents. You can also use the fact that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: What is the relationship between radical expressions and absolute value?

A: Radical expressions and absolute value are related in that they can be used to represent the same mathematical concept. For example, the expression $\sqrt{a}$ can be rewritten as $|a|^{\frac{1}{2}}$, and the expression $\sqrt[3]{a}$ can be rewritten as $|a|^{\frac{1}{3}}$.

Q: How do I simplify a radical expression that contains an absolute value?

A: To simplify a radical expression that contains an absolute value, you can use the property of absolute value that allows you to rewrite an absolute value expression as a square root expression. For example, the expression $\sqrt{|a|}$ can be rewritten as $|a|^{\frac{1}{2}}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying radical expressions. We have covered topics such as the definition of a radical expression, how to simplify a radical expression, and the relationship between radical expressions and fractional exponents. We hope that this article has been helpful in answering your questions and providing you with a better understanding of radical expressions.