Simplify The Expression: $ -\frac{3}{7} \left(-\frac{1}{9}\right) $

Feb 27, 2025 by ADMIN 68 views

Simplify the Expression: $ -\frac{3}{7} \left(-\frac{1}{9}\right) $

Understanding the Problem

When dealing with mathematical expressions, it's essential to understand the rules of operations and how to simplify them. In this case, we're given the expression $ -\frac{3}{7} \left(-\frac{1}{9}\right) $, and we need to simplify it. To do this, we'll apply the rules of multiplication and the properties of negative numbers.

The Rules of Multiplication

When multiplying two fractions, we multiply the numerators and denominators separately. In this case, we have:

$ -\frac{3}{7} \left(-\frac{1}{9}\right) = \frac{(-3)(-1)}{(7)(9)} $

The Properties of Negative Numbers

When multiplying two negative numbers, the result is always positive. This is because the negative signs cancel each other out. In this case, we have:

$ (-3)(-1) = 3 $

Simplifying the Expression

Now that we've applied the rules of multiplication and the properties of negative numbers, we can simplify the expression:

$ \frac{(-3)(-1)}{(7)(9)} = \frac{3}{63} $

Reducing the Fraction

To reduce the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 3 and 63 is 3. We can divide both the numerator and denominator by 3 to get:

$ \frac{3}{63} = \frac{1}{21} $

Conclusion

In conclusion, the simplified expression is $ \frac{1}{21} $. We applied the rules of multiplication and the properties of negative numbers to simplify the expression. By understanding the rules of operations and the properties of negative numbers, we can simplify complex mathematical expressions.

Additional Tips and Tricks

When multiplying two fractions, multiply the numerators and denominators separately.
When multiplying two negative numbers, the result is always positive.
To reduce a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in physics, we often need to simplify complex expressions to solve problems. In engineering, we use mathematical expressions to design and optimize systems. By simplifying expressions, we can make complex problems more manageable and easier to solve.

Common Mistakes to Avoid

When multiplying two fractions, don't forget to multiply the numerators and denominators separately.
When multiplying two negative numbers, don't forget that the result is always positive.
When reducing a fraction, don't forget to find the greatest common divisor (GCD) of the numerator and denominator.

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics, and it has many real-world applications. By understanding the rules of operations and the properties of negative numbers, we can simplify complex mathematical expressions. Remember to apply the rules of multiplication and the properties of negative numbers, and don't forget to reduce fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
Simplify the Expression: $ -\frac{3}{7} \left(-\frac{1}{9}\right) $ - Q&A

Understanding the Problem

Q&A

Q: What are the rules of multiplication when dealing with fractions?

A: When multiplying two fractions, we multiply the numerators and denominators separately. In this case, we have:

$ -\frac{3}{7} \left(-\frac{1}{9}\right) = \frac{(-3)(-1)}{(7)(9)} $

Q: What happens when we multiply two negative numbers?

A: When multiplying two negative numbers, the result is always positive. This is because the negative signs cancel each other out. In this case, we have:

$ (-3)(-1) = 3 $

Q: How do we simplify the expression?

A: Now that we've applied the rules of multiplication and the properties of negative numbers, we can simplify the expression:

$ \frac{(-3)(-1)}{(7)(9)} = \frac{3}{63} $

Q: How do we reduce the fraction?

A: To reduce the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 3 and 63 is 3. We can divide both the numerator and denominator by 3 to get:

$ \frac{3}{63} = \frac{1}{21} $

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

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

Forgetting to multiply the numerators and denominators separately when multiplying two fractions.
Forgetting that the result of multiplying two negative numbers is always positive.
Forgetting to find the greatest common divisor (GCD) of the numerator and denominator when reducing a fraction.

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

A: Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in physics, we often need to simplify complex expressions to solve problems. In engineering, we use mathematical expressions to design and optimize systems. By simplifying expressions, we can make complex problems more manageable and easier to solve.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through problems and exercises in your math textbook or online resources. You can also try simplifying expressions on your own by using real-world examples or creating your own problems.

Conclusion

Additional Resources

Khan Academy: Simplifying Expressions
Mathway: Simplifying Expressions
Wolfram Alpha: Simplifying Expressions

Practice Problems

Simplify the expression: $ -\frac{2}{3} \left(-\frac{4}{5}\right) $
Simplify the expression: $ \frac{1}{2} \left(-\frac{3}{4}\right) $
Simplify the expression: $ -\frac{5}{6} \left(-\frac{2}{3}\right) $

Answer Key

$ \frac{8}{15} $
$ -\frac{3}{8} $
$ \frac{10}{18} $