What Is The Fifth Term In The Expansion Of $(3x - 3y)^7$? □ X 3 Y 4 \square X^3 Y^4 □ X 3 Y 4

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What is the Fifth Term in the Expansion of (3x - 3y)^7?

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n, where a and b are numbers or variables and n is a positive integer. The theorem states that the expansion of (a + b)^n is given by:

(a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k

where the sum is taken over all non-negative integers k such that 0 ≤ k ≤ n.

In this problem, we are given the expression (3x - 3y)^7 and we are asked to find the fifth term in its expansion. To do this, we can use the binomial theorem with a = 3x and b = -3y.

The Binomial Theorem

The binomial theorem states that the expansion of (a + b)^n is given by:

(a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k

where the sum is taken over all non-negative integers k such that 0 ≤ k ≤ n.

Applying the Binomial Theorem

In this problem, we have a = 3x and b = -3y, and n = 7. We can plug these values into the formula for the binomial theorem to get:

(3x - 3y)^7 = ∑[7! / (k!(7-k)!)] * (3x)^(7-k) * (-3y)^k

Finding the Fifth Term

To find the fifth term in the expansion, we need to find the value of k such that k = 4. This is because the fifth term is the term where k = 4.

Calculating the Fifth Term

To calculate the fifth term, we need to plug k = 4 into the formula for the binomial theorem:

(3x - 3y)^7 = ∑[7! / (4!(7-4)!)] * (3x)^(7-4) * (-3y)^4

Simplifying the Expression

We can simplify the expression by calculating the factorials and the powers of x and y:

(3x - 3y)^7 = [7! / (4!3!)] * (3x)^3 * (-3y)^4

Evaluating the Factorials

We can evaluate the factorials as follows:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040 4! = 4 * 3 * 2 * 1 = 24 3! = 3 * 2 * 1 = 6

Substituting the Factorial Values

We can substitute the factorial values into the expression:

(3x - 3y)^7 = [5040 / (24 * 6)] * (3x)^3 * (-3y)^4

Simplifying the Expression

We can simplify the expression by canceling out the common factors:

(3x - 3y)^7 = [5040 / 144] * (3x)^3 * (-3y)^4

Evaluating the Expression

We can evaluate the expression as follows:

(3x - 3y)^7 = 35 * (3x)^3 * (-3y)^4

Simplifying the Expression

We can simplify the expression by evaluating the powers of x and y:

(3x - 3y)^7 = 35 * 27x^3 * 81y^4

Simplifying the Expression

We can simplify the expression by multiplying the numbers:

(3x - 3y)^7 = 28275x3y4

The Final Answer

Therefore, the fifth term in the expansion of (3x - 3y)^7 is 28275x3y4.

Conclusion

In this problem, we used the binomial theorem to expand the expression (3x - 3y)^7 and find the fifth term in its expansion. We applied the formula for the binomial theorem, evaluated the factorials, and simplified the expression to find the final answer.

Key Takeaways

  • The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n.
  • The expansion of (a + b)^n is given by the formula: (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k.
  • To find the fifth term in the expansion of (3x - 3y)^7, we need to plug k = 4 into the formula for the binomial theorem.
  • We can simplify the expression by evaluating the factorials and the powers of x and y.

References

  • "The Binomial Theorem" by Math Open Reference
  • "Binomial Theorem" by Wolfram MathWorld

Further Reading

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak

Related Topics

  • Binomial Theorem
  • Algebra
  • Calculus

Tags

  • Binomial Theorem
  • Algebra
  • Calculus
  • Math
  • Science
    Q&A: Binomial Theorem and Algebra

In this article, we will answer some common questions related to the binomial theorem and algebra.

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n, where a and b are numbers or variables and n is a positive integer.

Q: How do I apply the binomial theorem?

A: To apply the binomial theorem, you need to plug the values of a, b, and n into the formula: (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k.

Q: What is the formula for the binomial theorem?

A: The formula for the binomial theorem is: (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k.

Q: How do I find the fifth term in the expansion of (3x - 3y)^7?

A: To find the fifth term in the expansion of (3x - 3y)^7, you need to plug k = 4 into the formula for the binomial theorem and simplify the expression.

Q: What is the difference between the binomial theorem and the binomial expansion?

A: The binomial theorem is a formula that allows us to expand expressions of the form (a + b)^n, while the binomial expansion is the actual expansion of the expression.

Q: Can I use the binomial theorem to expand expressions with negative exponents?

A: Yes, you can use the binomial theorem to expand expressions with negative exponents. However, you need to be careful when simplifying the expression.

Q: How do I simplify the expression after applying the binomial theorem?

A: To simplify the expression after applying the binomial theorem, you need to evaluate the factorials and the powers of x and y.

Q: What are some common mistakes to avoid when applying the binomial theorem?

A: Some common mistakes to avoid when applying the binomial theorem include:

  • Not evaluating the factorials correctly
  • Not simplifying the expression correctly
  • Not using the correct formula for the binomial theorem

Q: Can I use the binomial theorem to solve problems in calculus?

A: Yes, you can use the binomial theorem to solve problems in calculus. However, you need to be careful when applying the formula and simplifying the expression.

Q: What are some real-world applications of the binomial theorem?

A: Some real-world applications of the binomial theorem include:

  • Probability theory
  • Statistics
  • Engineering
  • Computer science

Q: Can I use the binomial theorem to solve problems in physics?

A: Yes, you can use the binomial theorem to solve problems in physics. However, you need to be careful when applying the formula and simplifying the expression.

Q: What are some common misconceptions about the binomial theorem?

A: Some common misconceptions about the binomial theorem include:

  • The binomial theorem only applies to expressions with two terms
  • The binomial theorem only applies to positive integers
  • The binomial theorem is only used in algebra

Q: Can I use the binomial theorem to solve problems in other fields?

A: Yes, you can use the binomial theorem to solve problems in other fields, such as economics, finance, and biology.

Conclusion

In this article, we have answered some common questions related to the binomial theorem and algebra. We have also discussed some common mistakes to avoid and some real-world applications of the binomial theorem.

Key Takeaways

  • The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n.
  • The formula for the binomial theorem is: (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k.
  • To find the fifth term in the expansion of (3x - 3y)^7, you need to plug k = 4 into the formula for the binomial theorem and simplify the expression.
  • The binomial theorem has many real-world applications, including probability theory, statistics, engineering, and computer science.

References

  • "The Binomial Theorem" by Math Open Reference
  • "Binomial Theorem" by Wolfram MathWorld
  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak

Further Reading

  • "Probability Theory" by E.T. Jaynes
  • "Statistics" by David Freedman
  • "Engineering Mathematics" by K.A. Stroud
  • "Computer Science" by Robert Sedgewick

Related Topics

  • Binomial Theorem
  • Algebra
  • Calculus
  • Probability Theory
  • Statistics
  • Engineering
  • Computer Science

Tags

  • Binomial Theorem
  • Algebra
  • Calculus
  • Probability Theory
  • Statistics
  • Engineering
  • Computer Science
  • Math
  • Science