Evaluate The Expression:${ 27 \times \frac{2}{3} \div X \left( \frac{1}{8} \right)^{\frac{1}{3}} }$

Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying complex mathematical expressions to obtain a final value. This skill is essential in various mathematical operations, including algebra, geometry, and calculus. In this article, we will evaluate the expression: $27 \times \frac{2}{3} \div x \left( \frac{1}{8} \right)^{\frac{1}{3}}$. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final value.

Understanding the Expression

The given expression is a combination of multiplication, division, and exponentiation operations. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Breaking Down the Expression

Let's break down the expression into smaller parts:

1. $27 \times \frac{2}{3}$: This is a multiplication operation between two fractions.
2. $\div x$: This is a division operation between a fraction and a variable.
3. $\left( \frac{1}{8} \right)^{\frac{1}{3}}$: This is an exponential expression.

Simplifying the Multiplication Operation

To simplify the multiplication operation, we need to multiply the numerators and denominators separately:

$27 \times \frac{2}{3} = \frac{27 \times 2}{3}$

Using the commutative property of multiplication, we can rewrite the expression as:

$\frac{2 \times 27}{3}$

Now, we can simplify the expression by multiplying the numerators and denominators:

$\frac{2 \times 27}{3} = \frac{54}{3}$

Simplifying the Exponential Expression

To simplify the exponential expression, we need to raise the fraction to the power of $\frac{1}{3}$:

$\left( \frac{1}{8} \right)^{\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}}$

Using the property of exponents, we can rewrite the expression as:

$\frac{1}{\sqrt[3]{8}}$

Now, we can simplify the expression by evaluating the cube root:

$\frac{1}{\sqrt[3]{8}} = \frac{1}{2}$

Combining the Simplified Expressions

Now that we have simplified the multiplication and exponential operations, we can combine the simplified expressions:

$\frac{54}{3} \div x \left( \frac{1}{2} \right)$

Evaluating the Division Operation

To evaluate the division operation, we need to divide the fraction by the variable and the fraction:

$\frac{54}{3} \div x \left( \frac{1}{2} \right) = \frac{54}{3x} \times \frac{1}{2}$

Using the commutative property of multiplication, we can rewrite the expression as:

$\frac{54}{3x} \times \frac{1}{2} = \frac{54}{6x}$

Now, we can simplify the expression by dividing the numerator and denominator by their greatest common divisor:

$\frac{54}{6x} = \frac{9}{x}$

Conclusion

In conclusion, we have evaluated the expression: $27 \times \frac{2}{3} \div x \left( \frac{1}{8} \right)^{\frac{1}{3}}$. We broke down the expression into smaller parts, simplified each part, and then combined them to obtain the final value. The final value of the expression is $\frac{9}{x}$.

Final Answer

The final answer is $\boxed{\frac{9}{x}}$.

Frequently Asked Questions

• What is the order of operations in mathematics? The order of operations in mathematics is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• How do you simplify a multiplication operation between two fractions? To simplify a multiplication operation between two fractions, you need to multiply the numerators and denominators separately.
• How do you simplify an exponential expression? To simplify an exponential expression, you need to raise the fraction to the power of the exponent.
• How do you evaluate a division operation? To evaluate a division operation, you need to divide the fraction by the variable and the fraction.

References

• "Order of Operations" by Math Open Reference
• "Simplifying Fractions" by Math Is Fun
• "Exponents" by Khan Academy
• "Division" by Mathway
  

Introduction

Evaluating expressions is a crucial skill in mathematics that involves simplifying complex mathematical expressions to obtain a final value. In our previous article, we evaluated the expression: $27 \times \frac{2}{3} \div x \left( \frac{1}{8} \right)^{\frac{1}{3}}$. In this article, we will answer some frequently asked questions related to evaluating expressions.

Q&A

Q: What is the order of operations in mathematics?

A: The order of operations in mathematics is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do you simplify a multiplication operation between two fractions?

A: To simplify a multiplication operation between two fractions, you need to multiply the numerators and denominators separately. For example, $27 \times \frac{2}{3} = \frac{27 \times 2}{3}$.

Q: How do you simplify an exponential expression?

A: To simplify an exponential expression, you need to raise the fraction to the power of the exponent. For example, $\left( \frac{1}{8} \right)^{\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}}$.

Q: How do you evaluate a division operation?

A: To evaluate a division operation, you need to divide the fraction by the variable and the fraction. For example, $\frac{54}{3} \div x \left( \frac{1}{2} \right) = \frac{54}{3x} \times \frac{1}{2}$.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do you simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you need to find the least common multiple (LCM) of the denominator and the variable. For example, $\frac{1}{x} = \frac{1}{x} \times \frac{x}{x} = \frac{x}{x^2}$.

Q: How do you evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, you need to follow the order of operations (PEMDAS). For example, $3 \times 2 + 5 \div 2 = 6 + 2.5 = 8.5$.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two expressions are equal, while an expression is a mathematical statement that contains variables, constants, and mathematical operations.

Q: How do you solve an equation with a variable in the denominator?

A: To solve an equation with a variable in the denominator, you need to isolate the variable by multiplying both sides of the equation by the reciprocal of the denominator. For example, $\frac{x}{2} = 3 \Rightarrow x = 3 \times 2 = 6$.

Conclusion

In conclusion, evaluating expressions is a crucial skill in mathematics that involves simplifying complex mathematical expressions to obtain a final value. By following the order of operations (PEMDAS) and simplifying fractions and exponential expressions, you can evaluate expressions with ease. We hope this article has answered some of your frequently asked questions related to evaluating expressions.

Final Answer

The final answer is $\boxed{6}$.

Frequently Asked Questions

• What is the order of operations in mathematics?
• How do you simplify a multiplication operation between two fractions?
• How do you simplify an exponential expression?
• How do you evaluate a division operation?
• What is the difference between a variable and a constant?
• How do you simplify a fraction with a variable in the denominator?
• How do you evaluate an expression with multiple operations?
• What is the difference between an equation and an expression?
• How do you solve an equation with a variable in the denominator?

References

• "Order of Operations" by Math Open Reference
• "Simplifying Fractions" by Math Is Fun
• "Exponents" by Khan Academy
• "Division" by Mathway
• "Equations" by Mathway
• "Variables" by Khan Academy