Evaluate The Expression: { \left(\frac{4}{5}\right)^{-2}$}$

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Introduction

In mathematics, expressions involving exponents and fractions can be challenging to evaluate. The given expression, (45)−2\left(\frac{4}{5}\right)^{-2}, is a classic example of a problem that requires a deep understanding of exponent rules and fraction manipulation. In this article, we will delve into the world of exponents and fractions, and provide a step-by-step guide on how to evaluate the given expression.

Understanding Exponents and Fractions

Before we dive into the evaluation of the expression, it's essential to understand the basics of exponents and fractions. An exponent is a small number that is written above and to the right of a number or a variable, indicating how many times the base is multiplied by itself. For example, a2a^2 means aa multiplied by itself, or a×aa \times a. On the other hand, a fraction is a way of expressing a part of a whole, with a numerator and a denominator. For example, 45\frac{4}{5} means 4 parts out of 5.

Evaluating the Expression

To evaluate the expression (45)−2\left(\frac{4}{5}\right)^{-2}, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses, which is 45\frac{4}{5}.
  2. Exponents: Evaluate the exponent, which is −2-2.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 45\frac{4}{5}. This fraction is already simplified, so we can move on to the next step.

Step 2: Evaluate the Exponent

The exponent is −2-2. To evaluate this, we need to recall the rule for negative exponents: a−n=1ana^{-n} = \frac{1}{a^n}. In this case, we have (45)−2=1(45)2\left(\frac{4}{5}\right)^{-2} = \frac{1}{\left(\frac{4}{5}\right)^2}.

Step 3: Evaluate the Exponent of the Fraction

To evaluate the exponent of the fraction, we need to recall the rule for squaring a fraction: (ab)2=a2b2\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}. In this case, we have (45)2=4252=1625\left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25}.

Step 4: Substitute the Result Back into the Original Expression

Now that we have evaluated the exponent of the fraction, we can substitute the result back into the original expression: 1(45)2=11625\frac{1}{\left(\frac{4}{5}\right)^2} = \frac{1}{\frac{16}{25}}.

Step 5: Simplify the Expression

To simplify the expression, we need to recall the rule for dividing fractions: ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}. In this case, we have 11625=11625×2516=2516\frac{1}{\frac{16}{25}} = \frac{1}{\frac{16}{25}} \times \frac{25}{16} = \frac{25}{16}.

Conclusion

In conclusion, the expression (45)−2\left(\frac{4}{5}\right)^{-2} can be evaluated by following the order of operations (PEMDAS) and applying the rules for exponents and fractions. By simplifying the expression, we arrive at the final answer: 2516\frac{25}{16}.

Frequently Asked Questions

  • Q: What is the rule for negative exponents? A: The rule for negative exponents is a−n=1ana^{-n} = \frac{1}{a^n}.
  • Q: What is the rule for squaring a fraction? A: The rule for squaring a fraction is (ab)2=a2b2\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}.
  • Q: How do you simplify a fraction? A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD).

Final Answer

The final answer is: 2516\boxed{\frac{25}{16}}

Introduction

In our previous article, we evaluated the expression (45)−2\left(\frac{4}{5}\right)^{-2} using the order of operations (PEMDAS) and applying the rules for exponents and fractions. In this article, we will answer some frequently asked questions related to evaluating expressions with exponents and fractions.

Q&A

Q: What is the rule for negative exponents?

A: The rule for negative exponents is a−n=1ana^{-n} = \frac{1}{a^n}. This means that when you have a negative exponent, you can rewrite it as a fraction with the reciprocal of the base in the numerator and the exponent in the denominator.

Q: What is the rule for squaring a fraction?

A: The rule for squaring a fraction is (ab)2=a2b2\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}. This means that when you square a fraction, you can square the numerator and denominator separately.

Q: How do you simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD). This will give you the simplest form of the fraction.

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

A: A fraction is a way of expressing a part of a whole, with a numerator and a denominator. A decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 12\frac{1}{2} is equal to the decimal 0.5.

Q: How do you add and subtract fractions?

A: To add and subtract fractions, you need to have the same denominator. If the denominators are different, you need to find the least common multiple (LCM) of the denominators and convert both fractions to have that LCM as the denominator.

Q: How do you multiply and divide fractions?

A: To multiply and divide fractions, you can multiply and divide the numerators and denominators separately. For example, to multiply 12\frac{1}{2} and 34\frac{3}{4}, you can multiply the numerators (1 and 3) and denominators (2 and 4) separately to get 38\frac{3}{8}.

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

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

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

Conclusion

In conclusion, evaluating expressions with exponents and fractions requires a deep understanding of the rules for exponents, fractions, and the order of operations (PEMDAS). By following these rules and applying them to different types of expressions, you can simplify complex expressions and arrive at the final answer.

Final Answer

The final answer is: 2516\boxed{\frac{25}{16}}