Expanding the Cubes and Fourth Powers

Oct 13, 2025 by talynt.org 34 views

Introduction

In this article, we will simplify the given expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ . This involves expanding the cubes and fourth powers, and then combining like terms to obtain the final result. We will use the binomial theorem to expand the expressions and then simplify the resulting terms.

Expanding the Cubes and Fourth Powers

To simplify the given expression, we first need to expand the cubes and fourth powers. We can use the binomial theorem to do this.

Expanding $(1+\sqrt{2})^3$

The binomial theorem states that for any positive integer $n$ , we have:

(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n

Using this formula, we can expand $(1+\sqrt{2})^3$ as follows:

(1+\sqrt{2})^3 = \binom{3}{0} 1^3 (\sqrt{2})^0 + \binom{3}{1} 1^2 (\sqrt{2})^1 + \binom{3}{2} 1^1 (\sqrt{2})^2 + \binom{3}{3} 1^0 (\sqrt{2})^3

Simplifying this expression, we get:

(1+\sqrt{2})^3 = 1 + 3\sqrt{2} + 3(\sqrt{2})^2 + (\sqrt{2})^3

(1+\sqrt{2})^3 = 1 + 3\sqrt{2} + 6 + 2\sqrt{2}

(1+\sqrt{2})^3 = 7 + 5\sqrt{2}

Expanding $(2-\sqrt{3})^4$

Using the same formula, we can expand $(2-\sqrt{3})^4$ as follows:

(2-\sqrt{3})^4 = \binom{4}{0} 2^4 (-\sqrt{3})^0 + \binom{4}{1} 2^3 (-\sqrt{3})^1 + \binom{4}{2} 2^2 (-\sqrt{3})^2 + \binom{4}{3} 2^1 (-\sqrt{3})^3 + \binom{4}{4} 2^0 (-\sqrt{3})^4

Simplifying this expression, we get:

(2-\sqrt{3})^4 = 16 - 32\sqrt{3} + 24(\sqrt{3})^2 - 8(\sqrt{3})^3 + (\sqrt{3})^4

(2-\sqrt{3})^4 = 16 - 32\sqrt{3} + 24(3) - 8(3\sqrt{3}) + 9

(2-\sqrt{3})^4 = 16 - 32\sqrt{3} + 72 - 24\sqrt{3} + 9

(2-\sqrt{3})^4 = 97 - 56\sqrt{3}

Combining Like Terms

Now that we have expanded the cubes and fourth powers, we can combine like terms to simplify the expression.

We have:

(1+\sqrt{2})^3 = 7 + 5\sqrt{2}

(2-\sqrt{3})^4 = 97 - 56\sqrt{3}

Adding these two expressions together, we get:

(1+\sqrt{2})^3 + (2-\sqrt{3})^4 = (7 + 5\sqrt{2}) + (97 - 56\sqrt{3})

Combining like terms, we get:

(1+\sqrt{2})^3 + (2-\sqrt{3})^4 = 104 - 56\sqrt{3} + 5\sqrt{2}

Conclusion

In this article, we simplified the given expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ by expanding the cubes and fourth powers using the binomial theorem, and then combining like terms to obtain the final result. The final simplified expression is $104 - 56\sqrt{3} + 5\sqrt{2}$ .

References

Binomial Theorem: https://en.wikipedia.org/wiki/Binomial_theorem
Algebraic Manipulation: https://en.wikipedia.org/wiki/Algebraic_manipulation

Introduction

In our previous article, we simplified the given expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ by expanding the cubes and fourth powers using the binomial theorem, and then combining like terms to obtain the final result. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the binomial theorem and how is it used in simplifying algebraic expressions?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial raised to a power. It is used to simplify algebraic expressions by expanding the terms and then combining like terms. In the case of the expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ , we used the binomial theorem to expand the cubes and fourth powers.

Q: How do you expand a cube using the binomial theorem?

A: To expand a cube using the binomial theorem, you need to use the formula:

(a+b)^3 = \binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a b^2 + \binom{3}{3} a^0 b^3

You can then simplify the expression by combining like terms.

Q: How do you expand a fourth power using the binomial theorem?

A: To expand a fourth power using the binomial theorem, you need to use the formula:

(a+b)^4 = \binom{4}{0} a^4 b^0 + \binom{4}{1} a^3 b^1 + \binom{4}{2} a^2 b^2 + \binom{4}{3} a b^3 + \binom{4}{4} a^0 b^4

You can then simplify the expression by combining like terms.

Q: What is the difference between a cube and a fourth power?

A: A cube is a three-dimensional shape with six square faces, while a fourth power is a mathematical expression that is raised to the power of 4. In the context of the binomial theorem, a cube is expanded using the formula $(a+b)^3$ , while a fourth power is expanded using the formula $(a+b)^4$ .

Q: How do you combine like terms in an algebraic expression?

A: To combine like terms in an algebraic expression, you need to identify the terms that have the same variable and coefficient, and then add or subtract them. For example, in the expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ , we combined the like terms $5\sqrt{2}$ and $-56\sqrt{3}$ to obtain the final result.

Q: What is the final simplified expression for $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ ?

A: The final simplified expression for $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ is $104 - 56\sqrt{3} + 5\sqrt{2}$ .

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $(1+\sqrt{2})^3 + (2-\sqrt{3})^4$ . We hope that this article has provided you with a better understanding of the binomial theorem and how it is used in simplifying algebraic expressions.

References

Binomial Theorem: https://en.wikipedia.org/wiki/Binomial_theorem
Algebraic Manipulation: https://en.wikipedia.org/wiki/Algebraic_manipulation