A) 4 7 ÷ 1 3 4 \frac{4}{7} \div 1 \frac{3}{4} 7 4 ÷ 1 4 3 B) 5 3 ÷ 2 2 3 5^3 \div 2 \frac{2}{3} 5 3 ÷ 2 3 2

Mar 9, 2025 by ADMIN 114 views

**Dividing Mixed Numbers and Exponents: A Comprehensive Guide**

Introduction

In mathematics, division is a fundamental operation that involves finding the quotient of two numbers. When dealing with mixed numbers and exponents, division can become a bit more complex. In this article, we will explore the process of dividing mixed numbers and exponents, and provide step-by-step examples to help you understand the concept.

Dividing Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, 1 3/4 is a mixed number that consists of a whole number (1) and a fraction (3/4). When dividing mixed numbers, we need to follow a specific procedure to ensure that we get the correct result.

Step 1: Convert the Mixed Number to an Improper Fraction

To divide mixed numbers, we need to convert them to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

For example, let's convert 1 3/4 to an improper fraction:

1 3/4 = (1 × 4) + 3/4 = 4 + 3/4 = 19/4

Step 2: Invert the Divisor and Multiply

Once we have converted the mixed number to an improper fraction, we can proceed with the division. To divide an improper fraction by a mixed number, we need to invert the divisor (i.e., flip the numerator and denominator) and multiply.

For example, let's divide 4/7 by 1 3/4:

4/7 ÷ 1 3/4 = 4/7 ÷ 19/4

Invert the divisor: 19/4 becomes 4/19

Multiply: 4/7 × 4/19 = 16/133

Step 3: Simplify the Result

After multiplying, we need to simplify the result to its simplest form. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

For example, let's simplify 16/133:

GCD(16, 133) = 1

Since the GCD is 1, the fraction 16/133 is already in its simplest form.

Dividing Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, 5^3 means 5 × 5 × 5. When dividing exponents, we need to follow a specific rule to ensure that we get the correct result.

Rule 1: Divide the Coefficients

When dividing exponents, we need to divide the coefficients (i.e., the numbers in front of the exponents). For example, let's divide 5^3 by 2 2/3:

5^3 ÷ 2 2/3 = 5^3 ÷ 8/3

Divide the coefficients: 5 ÷ 8 = 5/8

Rule 2: Subtract the Exponents

Once we have divided the coefficients, we need to subtract the exponents. For example, let's subtract the exponents in 5^3 ÷ 8/3:

5^3 ÷ 8/3 = (5/8) × (3-3) = (5/8) × 0 = 0

Rule 3: Simplify the Result

After subtracting the exponents, we need to simplify the result to its simplest form. In this case, the result is 0, which is already in its simplest form.

Conclusion

Dividing mixed numbers and exponents can be a bit complex, but by following the steps outlined in this article, you can ensure that you get the correct result. Remember to convert mixed numbers to improper fractions, invert the divisor and multiply, and simplify the result to its simplest form. With practice and patience, you will become proficient in dividing mixed numbers and exponents.

Examples

Example 1: Dividing Mixed Numbers

Divide 4/7 by 1 3/4:

4/7 ÷ 1 3/4 = 4/7 ÷ 19/4

Invert the divisor: 19/4 becomes 4/19

Multiply: 4/7 × 4/19 = 16/133

Simplify the result: 16/133 is already in its simplest form.

Example 2: Dividing Exponents

Divide 5^3 by 2 2/3:

5^3 ÷ 2 2/3 = 5^3 ÷ 8/3

Divide the coefficients: 5 ÷ 8 = 5/8

Subtract the exponents: (5/8) × (3-3) = (5/8) × 0 = 0

Simplify the result: 0 is already in its simplest form.

Practice Problems

Divide 3/4 by 1 1/2.
Divide 2^4 by 3 1/3.
Divide 5^2 by 2 2/5.

Answer Key

3/4 ÷ 1 1/2 = 3/4 ÷ 3/2 = 1/2
2^4 ÷ 3 1/3 = 2^4 ÷ 10/3 = (16/3) × (3/10) = 16/30 = 8/15
5^2 ÷ 2 2/5 = 5^2 ÷ 12/5 = (25/5) × (5/12) = 25/60 = 5/12
Dividing Mixed Numbers and Exponents: A Comprehensive Guide ===========================================================

Q&A: Dividing Mixed Numbers and Exponents

Q: What is the difference between a mixed number and an improper fraction? A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.

Q: What is the correct order of operations when dividing mixed numbers? A: The correct order of operations is:

Convert the mixed number to an improper fraction
Invert the divisor and multiply
Simplify the result to its simplest form

Q: How do I divide exponents? A: To divide exponents, follow these steps:

Divide the coefficients (i.e., the numbers in front of the exponents)
Subtract the exponents
Simplify the result to its simplest form

Q: What is the rule for dividing exponents with the same base? A: When dividing exponents with the same base, subtract the exponents.

Q: Can I divide exponents with different bases? A: Yes, you can divide exponents with different bases. However, you need to follow the rule for dividing fractions, which is to invert the divisor and multiply.

Q: How do I simplify the result of dividing exponents? A: To simplify the result of dividing exponents, find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

Q: What is the difference between dividing mixed numbers and dividing exponents? A: Dividing mixed numbers involves converting the mixed number to an improper fraction, inverting the divisor, and multiplying, while dividing exponents involves dividing the coefficients and subtracting the exponents.

Q: Can I use a calculator to divide mixed numbers and exponents? A: Yes, you can use a calculator to divide mixed numbers and exponents. However, it's always a good idea to understand the concept and follow the steps outlined in this article to ensure accuracy.

Q: How do I practice dividing mixed numbers and exponents? A: You can practice dividing mixed numbers and exponents by working through the examples and practice problems provided in this article. You can also try creating your own problems and solving them to reinforce your understanding.

Q: What are some common mistakes to avoid when dividing mixed numbers and exponents? A: Some common mistakes to avoid when dividing mixed numbers and exponents include:

Not converting the mixed number to an improper fraction
Not inverting the divisor when dividing exponents
Not simplifying the result to its simplest form
Not following the correct order of operations

Q: How do I know if I'm doing the problem correctly? A: To ensure that you're doing the problem correctly, follow the steps outlined in this article and check your work by simplifying the result to its simplest form.