Question 1Simplify: 21 A 6 B 9 C 7 3 A 5 B 8 C 4 \frac{21 A^6 B^9 C^7}{3 A^5 B^8 C^4} 3 A 5 B 8 C 4 21 A 6 B 9 C 7 ​ Instructions:1. Use ∧ ^{\wedge} ∧ For Exponents. For Example, To Type X 2 X^2 X 2 , Type x^{\wedge}2.2. For Fractions, If There Is More Than One Term In The Numerator Or

Understanding the Basics of Exponents and Fractions

In mathematics, exponents and fractions are fundamental concepts that are used to represent complex expressions. Exponents are used to denote the power to which a number is raised, while fractions are used to represent a part of a whole. In this article, we will focus on simplifying complex fractions, specifically the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $a^3$ can be read as "a to the power of 3" and is equivalent to $a \times a \times a$. Exponents are used to simplify complex expressions and make them easier to work with.

What are Fractions?

Fractions are a way of representing a part of a whole. A fraction consists of two parts: the numerator and the denominator. The numerator represents the number of equal parts, while the denominator represents the total number of parts. For example, the fraction $\frac{1}{2}$ represents one half of a whole.

Simplifying Complex Fractions

Simplifying complex fractions involves reducing the fraction to its simplest form by canceling out any common factors in the numerator and denominator. To simplify the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$, we need to follow these steps:

Step 1: Factorize the Numerator and Denominator

The first step in simplifying the fraction is to factorize the numerator and denominator. The numerator can be factorized as $21 a^6 b^9 c^7 = 3 \times 7 \times a^6 \times b^9 \times c^7$, while the denominator can be factorized as $3 a^5 b^8 c^4 = 3 \times a^5 \times b^8 \times c^4$.

Step 2: Cancel Out Common Factors

Once we have factorized the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor of 3 in the numerator and denominator.

Step 3: Simplify the Remaining Expression

After canceling out the common factor, we are left with the expression $\frac{7 a^6 b^9 c^7}{a^5 b^8 c^4}$. To simplify this expression, we need to use the rule of exponents, which states that when we divide two powers with the same base, we subtract the exponents.

Step 4: Apply the Rule of Exponents

Using the rule of exponents, we can simplify the expression $\frac{7 a^6 b^9 c^7}{a^5 b^8 c^4}$ as follows:

$\frac{7 a^6 b^9 c^7}{a^5 b^8 c^4} = 7 a^{6-5} b^{9-8} c^{7-4}$

Step 5: Simplify the Expression

Simplifying the expression further, we get:

$7 a^{6-5} b^{9-8} c^{7-4} = 7 a^1 b^1 c^3$

Step 6: Write the Final Answer

Therefore, the simplified form of the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$ is $7 a b c^3$.

Conclusion

Simplifying complex fractions involves reducing the fraction to its simplest form by canceling out any common factors in the numerator and denominator. By following the steps outlined in this article, we can simplify the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$ to its simplest form, which is $7 a b c^3$. This demonstrates the importance of understanding exponents and fractions in mathematics and how they can be used to simplify complex expressions.

Final Answer

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in the numerator or denominator. For example, the fraction $\frac{1}{2} + \frac{1}{3}$ is a complex fraction.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

1. Factorize the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the remaining expression using the rule of exponents.
4. Write the final answer.

Q: What is the rule of exponents?

A: The rule of exponents states that when we divide two powers with the same base, we subtract the exponents. For example, $a^m \div a^n = a^{m-n}$.

Q: How do I apply the rule of exponents?

A: To apply the rule of exponents, you need to follow these steps:

1. Identify the base and the exponents in the numerator and denominator.
2. Subtract the exponents in the denominator from the exponents in the numerator.
3. Write the final answer.

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

A: A fraction is a simple fraction that contains only one numerator and one denominator. A complex fraction, on the other hand, is a fraction that contains one or more fractions in the numerator or denominator.

Q: How do I simplify a fraction with a variable in the numerator or denominator?

A: To simplify a fraction with a variable in the numerator or denominator, you need to follow these steps:

1. Factorize the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the remaining expression using the rule of exponents.
4. Write the final answer.

Q: What is the final answer to the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$?

A: The final answer to the fraction $\frac{21 a^6 b^9 c^7}{3 a^5 b^8 c^4}$ is $7 a b c^3$.

Q: How do I check my answer?

A: To check your answer, you need to follow these steps:

1. Multiply the numerator and denominator by the reciprocal of the denominator.
2. Simplify the expression using the rule of exponents.
3. Write the final answer.

Conclusion

Simplifying complex fractions involves reducing the fraction to its simplest form by canceling out any common factors in the numerator and denominator. By following the steps outlined in this article, you can simplify complex fractions and check your answers.

Final Answer

The final answer is: $\boxed{7 a b c^3}$