Evaluate The Expression:$ \left(-\frac{3}{4}\right)^2 + \frac{1}{4} \div -\frac{4}{5} $Write Your Answer As A Fraction, Whole Number, Or Mixed Number.

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Introduction

In this article, we will evaluate the given mathematical expression: (−34)2+14÷−45\left(-\frac{3}{4}\right)^2 + \frac{1}{4} \div -\frac{4}{5}. We will break down the expression into smaller parts, apply the order of operations, and simplify the result to obtain the final answer.

Understanding the Order of Operations

Before we start evaluating the expression, it's essential to understand the order of operations. The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  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.

Evaluating the Expression

Now that we understand the order of operations, let's evaluate the given expression:

(−34)2+14÷−45\left(-\frac{3}{4}\right)^2 + \frac{1}{4} \div -\frac{4}{5}

Step 1: Evaluate the Exponent

The first step is to evaluate the exponent. We have (−34)2\left(-\frac{3}{4}\right)^2. To evaluate this expression, we need to square the fraction:

(−34)2=916\left(-\frac{3}{4}\right)^2 = \frac{9}{16}

Step 2: Evaluate the Division

Next, we need to evaluate the division operation: 14÷−45\frac{1}{4} \div -\frac{4}{5}. To divide fractions, we need to invert the second fraction and multiply:

14÷−45=14×−54=−516\frac{1}{4} \div -\frac{4}{5} = \frac{1}{4} \times -\frac{5}{4} = -\frac{5}{16}

Step 3: Add the Results

Now that we have evaluated the exponent and the division operation, we can add the results:

916+(−516)\frac{9}{16} + \left(-\frac{5}{16}\right)

To add fractions, we need to have the same denominator. In this case, the denominator is 16, so we can add the fractions directly:

916+(−516)=9−516=416\frac{9}{16} + \left(-\frac{5}{16}\right) = \frac{9-5}{16} = \frac{4}{16}

Step 4: Simplify the Result

Finally, we can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 4 and 16 is 4, so we can simplify the fraction:

416=14\frac{4}{16} = \frac{1}{4}

Conclusion

In this article, we evaluated the given mathematical expression: (−34)2+14÷−45\left(-\frac{3}{4}\right)^2 + \frac{1}{4} \div -\frac{4}{5}. We broke down the expression into smaller parts, applied the order of operations, and simplified the result to obtain the final answer: 14\frac{1}{4}.

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations.

Q: How do I evaluate an exponent?

A: To evaluate an exponent, you need to raise the base to the power of the exponent. For example, (−34)2=916\left(-\frac{3}{4}\right)^2 = \frac{9}{16}.

Q: How do I evaluate a division operation?

A: To divide fractions, you need to invert the second fraction and multiply. For example, 14÷−45=14×−54=−516\frac{1}{4} \div -\frac{4}{5} = \frac{1}{4} \times -\frac{5}{4} = -\frac{5}{16}.

Q: How do I add fractions?

A: To add fractions, you need to have the same denominator. In this case, the denominator is 16, so we can add the fractions directly: 916+(−516)=9−516=416\frac{9}{16} + \left(-\frac{5}{16}\right) = \frac{9-5}{16} = \frac{4}{16}.

Final Answer

The final answer is: 14\boxed{\frac{1}{4}}

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Introduction

In our previous article, we evaluated the mathematical expression: (−34)2+14÷−45\left(-\frac{3}{4}\right)^2 + \frac{1}{4} \div -\frac{4}{5}. We broke down the expression into smaller parts, applied the order of operations, and simplified the result to obtain the final answer: 14\frac{1}{4}. In this article, we will answer some frequently asked questions related to evaluating mathematical expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations.

Q: How do I evaluate an exponent?

A: To evaluate an exponent, you need to raise the base to the power of the exponent. For example, (−34)2=916\left(-\frac{3}{4}\right)^2 = \frac{9}{16}.

Q: How do I evaluate a division operation?

A: To divide fractions, you need to invert the second fraction and multiply. For example, 14÷−45=14×−54=−516\frac{1}{4} \div -\frac{4}{5} = \frac{1}{4} \times -\frac{5}{4} = -\frac{5}{16}.

Q: How do I add fractions?

A: To add fractions, you need to have the same denominator. In this case, the denominator is 16, so we can add the fractions directly: 916+(−516)=9−516=416\frac{9}{16} + \left(-\frac{5}{16}\right) = \frac{9-5}{16} = \frac{4}{16}.

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

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, 12\frac{1}{2} is a fraction. A decimal is a way of expressing a number as a sum of powers of 10. For example, 0.5 is a decimal.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you need to divide the numerator by the denominator. For example, 12=0.5\frac{1}{2} = 0.5.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. For example, 0.5 = 12\frac{1}{2}.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a way of expressing a number as a combination of a whole number and a fraction. For example, 212\frac{1}{2} is a mixed number. An improper fraction is a way of expressing a number as a fraction with a numerator greater than the denominator. For example, 32\frac{3}{2} is an improper fraction.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you need to multiply the whole number by the denominator and add the numerator. For example, 212\frac{1}{2} = 52\frac{5}{2}.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the remainder as a fraction. For example, 52\frac{5}{2} = 212\frac{1}{2}.

Conclusion

In this article, we answered some frequently asked questions related to evaluating mathematical expressions. We covered topics such as the order of operations, evaluating exponents, division operations, adding fractions, converting fractions to decimals, converting decimals to fractions, mixed numbers, and improper fractions.

Final Answer

The final answer is: 14\boxed{\frac{1}{4}}