Write Down The Binomial Expansion Of \[$(x+2y)^5\$\].Use Your Expression To Evaluate \[$(t-0.2)^5\$\] And \[$(1.02)^5\$\].

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Introduction

The binomial expansion is a mathematical concept used to expand expressions of the form (a+b)n{(a+b)^n}, where a{a} and b{b} are constants or variables, and n{n} is a positive integer. In this article, we will focus on the binomial expansion of (x+2y)5{(x+2y)^5} and use it to evaluate two specific expressions: (tโˆ’0.2)5{(t-0.2)^5} and (1.02)5{(1.02)^5}.

Binomial Theorem

The binomial theorem is a formula that describes the expansion of (a+b)n{(a+b)^n}. It states that:

(a+b)n=(n0)an+(n1)anโˆ’1b+(n2)anโˆ’2b2+โ‹ฏ+(nnโˆ’1)abnโˆ’1+(nn)bn{(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n}

where (nk){\binom{n}{k}} is the binomial coefficient, defined as:

(nk)=n!k!(nโˆ’k)!{\binom{n}{k} = \frac{n!}{k!(n-k)!}}

Binomial Expansion of {(x+2y)^5$}$

Using the binomial theorem, we can expand (x+2y)5{(x+2y)^5} as follows:

(x+2y)5=(50)x5+(51)x4(2y)+(52)x3(2y)2+(53)x2(2y)3+(54)x(2y)4+(55)(2y)5{(x+2y)^5 = \binom{5}{0}x^5 + \binom{5}{1}x^4(2y) + \binom{5}{2}x^3(2y)^2 + \binom{5}{3}x^2(2y)^3 + \binom{5}{4}x(2y)^4 + \binom{5}{5}(2y)^5}

Expanding the binomial coefficients, we get:

(x+2y)5=x5+5x4(2y)+10x3(4y2)+10x2(8y3)+5x(16y4)+32y5{(x+2y)^5 = x^5 + 5x^4(2y) + 10x^3(4y^2) + 10x^2(8y^3) + 5x(16y^4) + 32y^5}

Simplifying the expression, we get:

(x+2y)5=x5+10x4y+40x3y2+80x2y3+80xy4+32y5{(x+2y)^5 = x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5}

Evaluating {(t-0.2)^5$}$

To evaluate (tโˆ’0.2)5{(t-0.2)^5}, we can substitute x=t{x = t} and y=โˆ’0.2{y = -0.2} into the expanded expression:

(tโˆ’0.2)5=t5+10t4(โˆ’0.2)+40t3(โˆ’0.2)2+80t2(โˆ’0.2)3+80t(โˆ’0.2)4+32(โˆ’0.2)5{(t-0.2)^5 = t^5 + 10t^4(-0.2) + 40t^3(-0.2)^2 + 80t^2(-0.2)^3 + 80t(-0.2)^4 + 32(-0.2)^5}

Simplifying the expression, we get:

(tโˆ’0.2)5=t5โˆ’2t4+1.6t3โˆ’0.64t2+0.128tโˆ’0.032{(t-0.2)^5 = t^5 - 2t^4 + 1.6t^3 - 0.64t^2 + 0.128t - 0.032}

Evaluating {(1.02)^5$}$

To evaluate (1.02)5{(1.02)^5}, we can substitute x=1.02{x = 1.02} and y=0{y = 0} into the expanded expression:

(1.02)5=(1.02)5+10(1.02)4(0)+40(1.02)3(0)2+80(1.02)2(0)3+80(1.02)(0)4+32(0)5{(1.02)^5 = (1.02)^5 + 10(1.02)^4(0) + 40(1.02)^3(0)^2 + 80(1.02)^2(0)^3 + 80(1.02)(0)^4 + 32(0)^5}

Simplifying the expression, we get:

(1.02)5=(1.02)5{(1.02)^5 = (1.02)^5}

Using a calculator, we can evaluate (1.02)5{(1.02)^5} as:

(1.02)5=1.104081984{(1.02)^5 = 1.104081984}

Conclusion

In this article, we have discussed the binomial expansion of (x+2y)5{(x+2y)^5} and used it to evaluate two specific expressions: (tโˆ’0.2)5{(t-0.2)^5} and (1.02)5{(1.02)^5}. The binomial expansion is a powerful tool for expanding expressions of the form (a+b)n{(a+b)^n}, and it has many applications in mathematics and other fields.

References

Further Reading

Introduction

In this article, we will answer some frequently asked questions about the binomial expansion. The binomial expansion is a mathematical concept used to expand expressions of the form (a+b)n{(a+b)^n}, where a{a} and b{b} are constants or variables, and n{n} is a positive integer.

Q: What is the binomial expansion?

A: The binomial expansion is a mathematical concept used to expand expressions of the form (a+b)n{(a+b)^n}, where a{a} and b{b} are constants or variables, and n{n} is a positive integer.

Q: How do I use the binomial expansion?

A: To use the binomial expansion, you need to follow these steps:

  1. Identify the values of a{a}, b{b}, and n{n}.
  2. Use the binomial theorem to expand the expression.
  3. Simplify the expression by combining like terms.

Q: What is the binomial theorem?

A: The binomial theorem is a formula that describes the expansion of (a+b)n{(a+b)^n}. It states that:

(a+b)n=(n0)an+(n1)anโˆ’1b+(n2)anโˆ’2b2+โ‹ฏ+(nnโˆ’1)abnโˆ’1+(nn)bn{(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n}

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the binomial theorem. It is defined as:

(nk)=n!k!(nโˆ’k)!{\binom{n}{k} = \frac{n!}{k!(n-k)!}}

Q: How do I evaluate the binomial expansion?

A: To evaluate the binomial expansion, you need to follow these steps:

  1. Identify the values of a{a}, b{b}, and n{n}.
  2. Use the binomial theorem to expand the expression.
  3. Simplify the expression by combining like terms.
  4. Evaluate the expression by substituting the values of a{a}, b{b}, and n{n}.

Q: Can I use the binomial expansion to evaluate expressions with negative exponents?

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

Q: Can I use the binomial expansion to evaluate expressions with fractional exponents?

A: Yes, you can use the binomial expansion to evaluate expressions with fractional exponents. However, you need to be careful when simplifying the expression.

Q: What are some common applications of the binomial expansion?

A: The binomial expansion has many applications in mathematics and other fields. Some common applications include:

  • Evaluating expressions with exponents
  • Simplifying expressions with multiple terms
  • Solving equations with exponents
  • Evaluating expressions with fractional exponents

Q: Can I use the binomial expansion to evaluate expressions with complex numbers?

A: Yes, you can use the binomial expansion to evaluate expressions with complex numbers. However, you need to be careful when simplifying the expression.

Q: Can I use the binomial expansion to evaluate expressions with matrices?

A: Yes, you can use the binomial expansion to evaluate expressions with matrices. However, you need to be careful when simplifying the expression.

Conclusion

In this article, we have answered some frequently asked questions about the binomial expansion. The binomial expansion is a powerful tool for expanding expressions of the form (a+b)n{(a+b)^n}, and it has many applications in mathematics and other fields.

References

Further Reading