Simplify The Following Expression:$\[ \frac{2}{9} \cdot \frac{14}{15} \cdot \left(-\frac{9}{10}\right) \\]

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Introduction

When it comes to simplifying complex mathematical expressions, one of the most common operations is multiplying fractions. In this article, we will delve into the world of fractions and explore the step-by-step process of simplifying the expression: 29⋅1415⋅(−910)\frac{2}{9} \cdot \frac{14}{15} \cdot \left(-\frac{9}{10}\right). By the end of this article, you will have a solid understanding of how to simplify complex fraction expressions and be able to apply this knowledge to a wide range of mathematical problems.

Understanding Fractions

Before we dive into the simplification process, let's take a moment to understand what fractions are and how they work. A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator (the top number) and a denominator (the bottom number). For example, the fraction 23\frac{2}{3} represents the number 2 divided by 3.

Multiplying Fractions

Now that we have a solid understanding of fractions, let's move on to the process of multiplying them. When multiplying fractions, we simply multiply the numerators together and the denominators together. This is represented by the following formula:

abâ‹…cd=aâ‹…cbâ‹…d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

For example, let's multiply the fractions 23\frac{2}{3} and 45\frac{4}{5}:

23â‹…45=2â‹…43â‹…5=815\frac{2}{3} \cdot \frac{4}{5} = \frac{2 \cdot 4}{3 \cdot 5} = \frac{8}{15}

Simplifying the Expression

Now that we have a solid understanding of how to multiply fractions, let's apply this knowledge to the expression we are trying to simplify: 29⋅1415⋅(−910)\frac{2}{9} \cdot \frac{14}{15} \cdot \left(-\frac{9}{10}\right). To simplify this expression, we will follow the same process we used in the previous example:

29⋅1415⋅(−910)=2⋅14⋅(−9)9⋅15⋅(−10)\frac{2}{9} \cdot \frac{14}{15} \cdot \left(-\frac{9}{10}\right) = \frac{2 \cdot 14 \cdot (-9)}{9 \cdot 15 \cdot (-10)}

Canceling Out Common Factors

Before we can simplify the expression further, we need to cancel out any common factors between the numerator and the denominator. In this case, we can cancel out the common factor of 9:

2⋅14⋅(−9)9⋅15⋅(−10)=2⋅14⋅(−1)15⋅(−10)\frac{2 \cdot 14 \cdot (-9)}{9 \cdot 15 \cdot (-10)} = \frac{2 \cdot 14 \cdot (-1)}{15 \cdot (-10)}

Simplifying the Expression Further

Now that we have canceled out the common factor of 9, we can simplify the expression further by multiplying the remaining numbers:

2⋅14⋅(−1)15⋅(−10)=−28−150\frac{2 \cdot 14 \cdot (-1)}{15 \cdot (-10)} = \frac{-28}{-150}

Final Simplification

The final step in simplifying the expression is to cancel out any remaining common factors between the numerator and the denominator. In this case, we can cancel out the common factor of 2:

−28−150=−14−75\frac{-28}{-150} = \frac{-14}{-75}

Conclusion

In conclusion, simplifying complex fraction expressions requires a solid understanding of fractions and the process of multiplying them. By following the step-by-step process outlined in this article, you will be able to simplify even the most complex fraction expressions with ease. Remember to always cancel out common factors between the numerator and the denominator, and to simplify the expression further by multiplying the remaining numbers.

Frequently Asked Questions

  • What is the process of multiplying fractions?
    • To multiply fractions, we simply multiply the numerators together and the denominators together.
  • How do I simplify a complex fraction expression?
    • To simplify a complex fraction expression, we need to cancel out any common factors between the numerator and the denominator, and then simplify the expression further by multiplying the remaining numbers.
  • What is the final simplified expression for the given problem?
    • The final simplified expression for the given problem is −14−75\frac{-14}{-75}.

Additional Resources

  • For more information on fractions and how to multiply them, check out the following resources:
    • Khan Academy: Fractions
    • Mathway: Multiplying Fractions
    • Wolfram Alpha: Fractions

Final Thoughts

Simplifying complex fraction expressions is a crucial skill to have in mathematics. By following the step-by-step process outlined in this article, you will be able to simplify even the most complex fraction expressions with ease. Remember to always cancel out common factors between the numerator and the denominator, and to simplify the expression further by multiplying the remaining numbers. With practice and patience, you will become a master of simplifying complex fraction expressions in no time.

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Q&A: Simplifying Complex Fraction Expressions

Q: What is the process of multiplying fractions?

A: To multiply fractions, we simply multiply the numerators together and the denominators together.

Example: 23â‹…45=2â‹…43â‹…5=815\frac{2}{3} \cdot \frac{4}{5} = \frac{2 \cdot 4}{3 \cdot 5} = \frac{8}{15}

Q: How do I simplify a complex fraction expression?

A: To simplify a complex fraction expression, we need to cancel out any common factors between the numerator and the denominator, and then simplify the expression further by multiplying the remaining numbers.

Example: 29⋅1415⋅(−910)=−28−150=−14−75\frac{2}{9} \cdot \frac{14}{15} \cdot \left(-\frac{9}{10}\right) = \frac{-28}{-150} = \frac{-14}{-75}

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is −14−75\frac{-14}{-75}.

Q: Can I simplify a fraction expression with negative numbers?

A: Yes, you can simplify a fraction expression with negative numbers. When multiplying fractions with negative numbers, we simply multiply the numerators together and the denominators together, just like with positive numbers.

Example: −23⋅45=−2⋅43⋅5=−815\frac{-2}{3} \cdot \frac{4}{5} = \frac{-2 \cdot 4}{3 \cdot 5} = \frac{-8}{15}

Q: How do I cancel out common factors between the numerator and the denominator?

A: To cancel out common factors between the numerator and the denominator, we need to find the greatest common factor (GCF) of the two numbers and divide both the numerator and the denominator by the GCF.

Example: 1218=6â‹…26â‹…3=23\frac{12}{18} = \frac{6 \cdot 2}{6 \cdot 3} = \frac{2}{3}

Q: What is the greatest common factor (GCF) of two numbers?

A: The greatest common factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder.

Example: The GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: Can I simplify a fraction expression with decimals?

A: Yes, you can simplify a fraction expression with decimals. When simplifying a fraction expression with decimals, we can convert the decimals to fractions and then simplify the expression.

Example: 0.50.25=1/21/4=21=2\frac{0.5}{0.25} = \frac{1/2}{1/4} = \frac{2}{1} = 2

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

A: To convert a decimal to a fraction, we can write the decimal as a fraction with a denominator of 10 or 100, and then simplify the fraction.

Example: 0.5=510=120.5 = \frac{5}{10} = \frac{1}{2}

Q: Can I simplify a fraction expression with variables?

A: Yes, you can simplify a fraction expression with variables. When simplifying a fraction expression with variables, we can cancel out any common factors between the numerator and the denominator, and then simplify the expression further by multiplying the remaining numbers.

Example: xyâ‹…zw=xâ‹…zyâ‹…w\frac{x}{y} \cdot \frac{z}{w} = \frac{x \cdot z}{y \cdot w}

Conclusion

Simplifying complex fraction expressions is a crucial skill to have in mathematics. By following the step-by-step process outlined in this article, you will be able to simplify even the most complex fraction expressions with ease. Remember to always cancel out common factors between the numerator and the denominator, and to simplify the expression further by multiplying the remaining numbers. With practice and patience, you will become a master of simplifying complex fraction expressions in no time.

Additional Resources

  • For more information on fractions and how to multiply them, check out the following resources:
    • Khan Academy: Fractions
    • Mathway: Multiplying Fractions
    • Wolfram Alpha: Fractions

Final Thoughts

Simplifying complex fraction expressions is a crucial skill to have in mathematics. By following the step-by-step process outlined in this article, you will be able to simplify even the most complex fraction expressions with ease. Remember to always cancel out common factors between the numerator and the denominator, and to simplify the expression further by multiplying the remaining numbers. With practice and patience, you will become a master of simplifying complex fraction expressions in no time.