Simplify The Expression: $\[ \frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} : 2 \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. One of the most common expressions we encounter is the fraction, and simplifying it can be a bit tricky. In this article, we will simplify the expression $\frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} : 2$ using the properties of exponents and fractions.

Understanding the Expression

The given expression is $\frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} : 2$. To simplify this expression, we need to understand the properties of exponents and fractions. The expression consists of two terms: $27^{\frac{1}{6}}$ and $27^{\frac{1}{2}}$. Both terms are raised to the power of $\frac{1}{6}$ and $\frac{1}{2}$, respectively.

Simplifying the Expression

To simplify the expression, we can use the property of exponents that states $a^m \div a^n = a^{m-n}$. We can rewrite the expression as $\frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} = 27^{\frac{1}{6} - \frac{1}{2}}$. Now, we can simplify the exponent by finding the common denominator, which is 6.

$\frac{1}{6} - \frac{1}{2} = \frac{1}{6} - \frac{3}{6} = -\frac{2}{6} = -\frac{1}{3}$

So, the expression becomes $27^{-\frac{1}{3}}$. Now, we can simplify the expression further by rewriting it as $\frac{1}{27^{\frac{1}{3}}}$.

Simplifying the Fraction

The expression $\frac{1}{27^{\frac{1}{3}}}$ can be simplified further by rewriting it as $\frac{1}{\sqrt[3]{27}}$. Now, we can simplify the expression by finding the cube root of 27, which is 3.

$\frac{1}{\sqrt[3]{27}} = \frac{1}{3}$

Conclusion

In this article, we simplified the expression $\frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} : 2$ using the properties of exponents and fractions. We rewrote the expression as $27^{\frac{1}{6} - \frac{1}{2}}$, simplified the exponent, and finally simplified the fraction to get the final answer, which is $\frac{1}{3}$.

Final Answer

The final answer is $\boxed{\frac{1}{3}}$.

Properties of Exponents

In mathematics, exponents are used to represent repeated multiplication. The property of exponents states that $a^m \div a^n = a^{m-n}$. This property can be used to simplify expressions with exponents.

Examples of Simplifying Expressions

Here are some examples of simplifying expressions using the property of exponents:

• $\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$
• $\frac{3^4}{3^3} = 3^{4-3} = 3^1 = 3$
• $\frac{4^2}{4^1} = 4^{2-1} = 4^1 = 4$

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Use the property of exponents to simplify expressions with exponents.
• Rewrite the expression as a fraction and simplify the numerator and denominator separately.
• Use the property of fractions to simplify the expression.

Conclusion

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Introduction

In our previous article, we simplified the expression $\frac{27^{\frac{1}{6}}}{27^{\frac{1}{2}}} : 2$ using the properties of exponents and fractions. In this article, we will provide a Q&A guide to help you understand the concept better.

Q: What is the property of exponents?

A: The property of exponents states that $a^m \div a^n = a^{m-n}$. This property can be used to simplify expressions with exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the property of exponents to rewrite the expression as a fraction and simplify the numerator and denominator separately.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a part of a whole using a decimal point. For example, the fraction $\frac{1}{2}$ is equal to the decimal 0.5.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can use the property of fractions to rewrite the fraction as a product of two numbers. For example, the fraction $\frac{2}{4}$ can be simplified to $\frac{1}{2}$.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because $3 \times 3 \times 3 = 27$.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can use the property of cube roots to rewrite the expression as a product of two numbers. For example, the expression $\sqrt[3]{27}$ can be simplified to 3.

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 a fraction, while an irrational number is a number that cannot be expressed as a fraction. For example, the number 3 is a rational number, while the number $\sqrt{2}$ is an irrational number.

Q: How do I simplify an expression with a rational number and an irrational number?

A: To simplify an expression with a rational number and an irrational number, you can use the property of rational numbers to rewrite the expression as a product of two numbers. For example, the expression $\frac{\sqrt{2}}{2}$ can be simplified to $\frac{1}{\sqrt{2}}$.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of simplifying expressions with exponents and fractions. We discussed the property of exponents, the difference between a fraction and a decimal, and how to simplify a fraction. We also discussed the cube root of a number, how to simplify an expression with a cube root, and the difference between a rational number and an irrational number. Finally, we provided some tips and tricks to help you simplify expressions.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Use the property of exponents to simplify expressions with exponents.
• Rewrite the expression as a fraction and simplify the numerator and denominator separately.
• Use the property of fractions to simplify the expression.
• Use the property of cube roots to simplify expressions with cube roots.
• Use the property of rational numbers to simplify expressions with rational numbers and irrational numbers.

Final Answer

The final answer is $\boxed{\frac{1}{3}}$.