Simplify The Expression:$\[ \left[ \left(-\frac{1}{2}\right) + \left(-\frac{1}{2} - \frac{2}{3} + \frac{2}{2}\right) \right] \div \left(-\frac{2}{3}\right)^2 \cdot \left(-\frac{2}{3}\right)^{-3} \cdot \left(\frac{2}{3}\right)^2 \\]

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Introduction

In this article, we will simplify the given mathematical expression step by step. The expression involves fractions, exponents, and basic arithmetic operations. Our goal is to simplify the expression to its simplest form.

The Given Expression

The given expression is:

[(βˆ’12)+(βˆ’12βˆ’23+22)]Γ·(βˆ’23)2β‹…(βˆ’23)βˆ’3β‹…(23)2\left[ \left(-\frac{1}{2}\right) + \left(-\frac{1}{2} - \frac{2}{3} + \frac{2}{2}\right) \right] \div \left(-\frac{2}{3}\right)^2 \cdot \left(-\frac{2}{3}\right)^{-3} \cdot \left(\frac{2}{3}\right)^2

Step 1: Simplify the Fractions Inside the Parentheses

Let's start by simplifying the fractions inside the parentheses.

(βˆ’12)+(βˆ’12βˆ’23+22)\left(-\frac{1}{2}\right) + \left(-\frac{1}{2} - \frac{2}{3} + \frac{2}{2}\right)

We can simplify the fractions by finding a common denominator.

(βˆ’12)+(βˆ’36βˆ’46+66)\left(-\frac{1}{2}\right) + \left(-\frac{3}{6} - \frac{4}{6} + \frac{6}{6}\right)

Now, we can add and subtract the fractions.

(βˆ’12)+(βˆ’36βˆ’46+66)=(βˆ’12)+(βˆ’16)\left(-\frac{1}{2}\right) + \left(-\frac{3}{6} - \frac{4}{6} + \frac{6}{6}\right) = \left(-\frac{1}{2}\right) + \left(-\frac{1}{6}\right)

Step 2: Simplify the Exponents

Next, let's simplify the exponents.

(βˆ’23)2β‹…(βˆ’23)βˆ’3β‹…(23)2\left(-\frac{2}{3}\right)^2 \cdot \left(-\frac{2}{3}\right)^{-3} \cdot \left(\frac{2}{3}\right)^2

We can simplify the exponents by applying the rules of exponents.

(βˆ’23)2=49\left(-\frac{2}{3}\right)^2 = \frac{4}{9}

(βˆ’23)βˆ’3=(32)3=278\left(-\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}

(23)2=49\left(\frac{2}{3}\right)^2 = \frac{4}{9}

Step 3: Multiply the Simplified Exponents

Now, let's multiply the simplified exponents.

49β‹…278β‹…49\frac{4}{9} \cdot \frac{27}{8} \cdot \frac{4}{9}

We can multiply the fractions by multiplying the numerators and denominators.

49β‹…278β‹…49=4β‹…27β‹…49β‹…8β‹…9\frac{4}{9} \cdot \frac{27}{8} \cdot \frac{4}{9} = \frac{4 \cdot 27 \cdot 4}{9 \cdot 8 \cdot 9}

Step 4: Simplify the Result

Now, let's simplify the result.

4β‹…27β‹…49β‹…8β‹…9=432648\frac{4 \cdot 27 \cdot 4}{9 \cdot 8 \cdot 9} = \frac{432}{648}

We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 72.

432648=69\frac{432}{648} = \frac{6}{9}

Step 5: Simplify the Final Expression

Now, let's simplify the final expression.

[(βˆ’12)+(βˆ’12βˆ’23+22)]Γ·(βˆ’23)2β‹…(βˆ’23)βˆ’3β‹…(23)2\left[ \left(-\frac{1}{2}\right) + \left(-\frac{1}{2} - \frac{2}{3} + \frac{2}{2}\right) \right] \div \left(-\frac{2}{3}\right)^2 \cdot \left(-\frac{2}{3}\right)^{-3} \cdot \left(\frac{2}{3}\right)^2

We can simplify the expression by substituting the simplified values.

[(βˆ’12)+(βˆ’16)]Γ·69\left[ \left(-\frac{1}{2}\right) + \left(-\frac{1}{6}\right) \right] \div \frac{6}{9}

We can simplify the expression by adding the fractions.

[(βˆ’12)+(βˆ’16)]Γ·69=(βˆ’36)Γ·69\left[ \left(-\frac{1}{2}\right) + \left(-\frac{1}{6}\right) \right] \div \frac{6}{9} = \left(-\frac{3}{6}\right) \div \frac{6}{9}

We can simplify the expression by dividing the fractions.

(βˆ’36)Γ·69=(βˆ’12)β‹…96\left(-\frac{3}{6}\right) \div \frac{6}{9} = \left(-\frac{1}{2}\right) \cdot \frac{9}{6}

We can simplify the expression by multiplying the fractions.

(βˆ’12)β‹…96=(βˆ’36)\left(-\frac{1}{2}\right) \cdot \frac{9}{6} = \left(-\frac{3}{6}\right)

The final answer is βˆ’12\boxed{-\frac{1}{2}}.

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Introduction

In our previous article, we simplified the given mathematical expression step by step. In this article, we will answer some frequently asked questions related to the expression and provide additional insights.

Q&A

Q: What is the main concept behind simplifying the expression?

A: The main concept behind simplifying the expression is to apply the rules of arithmetic operations, such as addition, subtraction, multiplication, and division, to simplify the expression.

Q: How do I simplify fractions inside parentheses?

A: To simplify fractions inside parentheses, you need to find a common denominator and then add or subtract the fractions.

Q: What is the rule for simplifying exponents?

A: The rule for simplifying exponents is to apply the power of a power rule, which states that (am)n = a^(m*n).

Q: How do I multiply fractions with exponents?

A: To multiply fractions with exponents, you need to multiply the numerators and denominators separately and then apply the exponent rule.

Q: What is the final answer to the expression?

A: The final answer to the expression is -\frac{1}{2}.

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by canceling out any common factors between the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

  • Not finding a common denominator when adding or subtracting fractions
  • Not applying the exponent rule correctly
  • Not multiplying the numerators and denominators separately when multiplying fractions with exponents
  • Not canceling out any common factors between the numerator and denominator

Additional Insights

Simplifying Expressions with Multiple Operations

When simplifying expressions with multiple operations, it's essential 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 any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying Expressions with Negative Numbers

When simplifying expressions with negative numbers, it's essential to remember that a negative number multiplied by a negative number is a positive number.

Simplifying Expressions with Fractions

When simplifying expressions with fractions, it's essential to find a common denominator and then add or subtract the fractions.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a deep understanding of arithmetic operations, exponents, and fractions. By following the rules and guidelines outlined in this article, you can simplify expressions with confidence and accuracy.

Final Tips

  • Always follow the order of operations (PEMDAS) when simplifying expressions.
  • Find a common denominator when adding or subtracting fractions.
  • Apply the exponent rule correctly when simplifying expressions with exponents.
  • Cancel out any common factors between the numerator and denominator.
  • Practice, practice, practice! The more you practice simplifying expressions, the more confident and accurate you will become.