( 2/3)^-3 × (3-1+6-1)^-2​

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Introduction

In mathematics, exponential expressions are a fundamental concept that plays a crucial role in various mathematical operations. When dealing with exponential expressions, it's essential to simplify them to make calculations easier and more manageable. In this article, we will explore how to simplify the expression (2/3)^-3 × (3-1+6-1)^-2, using step-by-step explanations and examples.

Understanding Exponents

Before we dive into simplifying the given expression, let's briefly review the concept of exponents. An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base number or variable is raised. For example, in the expression 2^3, the exponent 3 indicates that the base number 2 is raised to the power of 3.

Simplifying the First Term

The first term in the given expression is (2/3)^-3. To simplify this term, we need to apply the rule for negative exponents, which states that a^(-n) = 1/a^n. Applying this rule, we get:

(2/3)^-3 = 1 / (2/3)^3

Now, let's simplify the expression inside the parentheses:

(2/3)^3 = (2^3) / (3^3)

Using the rule for exponents, we can rewrite this expression as:

(2^3) / (3^3) = 8 / 27

Now, let's substitute this expression back into the original term:

(2/3)^-3 = 1 / (8 / 27)

To simplify this expression, we can multiply both the numerator and the denominator by 27:

(2/3)^-3 = (27 / 8)

Simplifying the Second Term

The second term in the given expression is (3-1+6-1)^-2. To simplify this term, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses: 3 - 1 = 2 6 - 1 = 5 2 + 5 = 7

  2. Raise 7 to the power of -2: 7^(-2) = 1 / 7^2

Now, let's simplify the expression inside the parentheses:

7^2 = 49

Now, let's substitute this expression back into the original term:

(3-1+6-1)^-2 = 1 / 49

Multiplying the Two Terms

Now that we have simplified both terms, we can multiply them together:

(2/3)^-3 × (3-1+6-1)^-2 = (27 / 8) × (1 / 49)

To multiply these fractions, we can multiply the numerators and the denominators separately:

(27 / 8) × (1 / 49) = (27 × 1) / (8 × 49)

Now, let's simplify the expression:

(27 × 1) = 27 (8 × 49) = 392

Now, let's substitute these expressions back into the original product:

(2/3)^-3 × (3-1+6-1)^-2 = 27 / 392

Conclusion

In this article, we have explored how to simplify the expression (2/3)^-3 × (3-1+6-1)^-2 using step-by-step explanations and examples. We have applied the rule for negative exponents and the order of operations to simplify the expression. The final simplified expression is 27 / 392.

Common Mistakes to Avoid

When simplifying exponential expressions, it's essential to avoid common mistakes. Here are a few common mistakes to watch out for:

  • Incorrect application of the rule for negative exponents: Make sure to apply the rule for negative exponents correctly. A^(-n) = 1/a^n.
  • Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Incorrect simplification of fractions: Make sure to simplify fractions correctly by multiplying the numerators and the denominators separately.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

  • Simplify the expression (4/5)^-2 × (2-1+3-1)^-3.
  • Simplify the expression (3/4)^-4 × (1-2+5-2)^-5.

Real-World Applications

Exponential expressions have numerous real-world applications. Here are a few examples:

  • Finance: Exponential expressions are used to calculate compound interest and investment returns.
  • Science: Exponential expressions are used to model population growth and decay.
  • Engineering: Exponential expressions are used to calculate the decay of radioactive materials.

Conclusion

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that a^(-n) = 1/a^n. This means that when a negative exponent is applied to a number or variable, we can rewrite it as a fraction with the number or variable in the denominator.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we need to apply the rule for negative exponents. We can rewrite the expression as a fraction with the number or variable in the denominator. For example, (2/3)^-3 can be rewritten as 1 / (2/3)^3.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

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

Q: How do I simplify an expression with multiple terms?

A: To simplify an expression with multiple terms, we need to follow the order of operations (PEMDAS). We can start by evaluating expressions inside parentheses, then evaluate exponents, and finally evaluate multiplication and division operations from left to right.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base number or variable is raised to a power, while a negative exponent indicates that the base number or variable is taken to a power. For example, 2^3 means 2 is raised to the power of 3, while 2^(-3) means 2 is taken to the power of -3.

Q: Can I simplify an expression with a variable exponent?

A: Yes, you can simplify an expression with a variable exponent. To do this, we need to apply the rule for negative exponents and the order of operations. For example, (2x)^-2 can be rewritten as 1 / (2x)^2.

Q: How do I simplify an expression with a fraction exponent?

A: To simplify an expression with a fraction exponent, we need to apply the rule for negative exponents and the order of operations. For example, (2/3)^(-2/3) can be rewritten as 1 / (2/3)^(2/3).

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

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

  • Incorrect application of the rule for negative exponents: Make sure to apply the rule for negative exponents correctly. A^(-n) = 1/a^n.
  • Incorrect order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Incorrect simplification of fractions: Make sure to simplify fractions correctly by multiplying the numerators and the denominators separately.

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through practice problems, such as:

  • Simplify the expression (4/5)^-2 × (2-1+3-1)^-3.
  • Simplify the expression (3/4)^-4 × (1-2+5-2)^-5.

Q: What are some real-world applications of exponential expressions?

A: Exponential expressions have numerous real-world applications, including:

  • Finance: Exponential expressions are used to calculate compound interest and investment returns.
  • Science: Exponential expressions are used to model population growth and decay.
  • Engineering: Exponential expressions are used to calculate the decay of radioactive materials.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill in mathematics. By applying the rule for negative exponents and the order of operations, we can simplify complex expressions and make calculations easier and more manageable. Remember to avoid common mistakes and practice simplifying exponential expressions to become proficient in this skill.