Expand The Expression $(x + 54)^5$ Without Simplifying Or Using A Straightforward Method.

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Introduction

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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 any real numbers and $n$ is a positive integer. In this article, we will use the binomial theorem to expand the expression $(x + 54)^5$ without simplifying or using a straightforward method.

The Binomial Theorem

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

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

where $\binom{n}{k}$ is the binomial coefficient, defined as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Expanding the Expression

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To expand the expression $(x + 54)^5$, we will use the binomial theorem with $a = x$, $b = 54$, and $n = 5$. Plugging these values into the formula, we get:

$$(x + 54)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} 54^k$$

Calculating the Binomial Coefficients

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To calculate the binomial coefficients, we will use the formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

We will calculate the binomial coefficients for each value of $k$ from $0$ to $5$.

Calculating $\binom{5}{0}$

$$\binom{5}{0} = \frac{5!}{0!(5-0)!} = \frac{5!}{0!5!} = 1$$

Calculating $\binom{5}{1}$

$$\binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{5!}{1!4!} = 5$$

Calculating $\binom{5}{2}$

$$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = 10$$

Calculating $\binom{5}{3}$

$$\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = 10$$

Calculating $\binom{5}{4}$

$$\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!} = 5$$

Calculating $\binom{5}{5}$

$$\binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{5!}{5!0!} = 1$$

Expanding the Expression

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Now that we have calculated the binomial coefficients, we can expand the expression $(x + 54)^5$:

$$(x + 54)^5 = \binom{5}{0} x^5 54^0 + \binom{5}{1} x^4 54^1 + \binom{5}{2} x^3 54^2 + \binom{5}{3} x^2 54^3 + \binom{5}{4} x^1 54^4 + \binom{5}{5} x^0 54^5$$

Simplifying the Expression

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We can simplify the expression by calculating the values of the binomial coefficients and the powers of $x$ and $54$:

$$(x + 54)^5 = 1 \cdot x^5 \cdot 54^0 + 5 \cdot x^4 \cdot 54^1 + 10 \cdot x^3 \cdot 54^2 + 10 \cdot x^2 \cdot 54^3 + 5 \cdot x^1 \cdot 54^4 + 1 \cdot x^0 \cdot 54^5$$

Calculating the Powers of $x$ and $54$

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We can calculate the powers of $x$ and $54$:

$$x^5 = x^5$$

$$54^0 = 1$$

$$54^1 = 54$$

$$54^2 = 2916$$

$$54^3 = 158184$$

$$54^4 = 8563716$$

$$54^5 = 464100624$$

Substituting the Values

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We can substitute the values of the powers of $x$ and $54$ into the expression:

$$(x + 54)^5 = x^5 + 5x^4 \cdot 54 + 10x^3 \cdot 2916 + 10x^2 \cdot 158184 + 5x^1 \cdot 8563716 + 1 \cdot 464100624$$

Simplifying the Expression

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We can simplify the expression by combining like terms:

$$(x + 54)^5 = x^5 + 270x^4 + 29160x^3 + 1581840x^2 + 42818580x + 464100624$$

Conclusion

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In this article, we used the binomial theorem to expand the expression $(x + 54)^5$ without simplifying or using a straightforward method. We calculated the binomial coefficients and the powers of $x$ and $54$, and then substituted the values into the expression. The final result is a polynomial of degree $5$.

References

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• [1] Binomial Theorem. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/wiki/Binomial_theorem
• [2] Binomial Coefficient. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/wiki/Binomial_coefficient

Note: The references provided are for informational purposes only and are not directly related to the content of this article.

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Introduction

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In our previous article, we used the binomial theorem to expand the expression $(x + 54)^5$ without simplifying or using a straightforward method. In this article, we will answer some common questions related to expanding the expression $(x + 54)^5$.

Q: What is the binomial theorem?

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A: The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a + b)^n$, where $a$ and $b$ are any real numbers and $n$ is a positive integer.

Q: How do I use the binomial theorem to expand the expression $(x + 54)^5$?

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A: To use the binomial theorem to expand the expression $(x + 54)^5$, you need to follow these steps:

1. Identify the values of $a$, $b$, and $n$ in the expression $(x + 54)^5$. In this case, $a = x$, $b = 54$, and $n = 5$.
2. Calculate the binomial coefficients using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Substitute the values of the binomial coefficients and the powers of $x$ and $54$ into the expression.

Q: What are the binomial coefficients?

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A: The binomial coefficients are the coefficients of the terms in the expanded expression. They are calculated using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Q: How do I calculate the powers of $x$ and $54$?

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A: To calculate the powers of $x$ and $54$, you need to follow these steps:

1. Identify the power of $x$ and $54$ in the term.
2. Calculate the value of the power using the formula $x^k = x \cdot x \cdot \ldots \cdot x$ (k times) and $54^k = 54 \cdot 54 \cdot \ldots \cdot 54$ (k times).

Q: What is the final result of expanding the expression $(x + 54)^5$?

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A: The final result of expanding the expression $(x + 54)^5$ is a polynomial of degree $5$:

$$(x + 54)^5 = x^5 + 270x^4 + 29160x^3 + 1581840x^2 + 42818580x + 464100624$$

Q: Can I simplify the expression $(x + 54)^5$?

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A: Yes, you can simplify the expression $(x + 54)^5$ by combining like terms.

Q: What are some common mistakes to avoid when expanding the expression $(x + 54)^5$?

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A: Some common mistakes to avoid when expanding the expression $(x + 54)^5$ include:

• Not using the binomial theorem correctly
• Not calculating the binomial coefficients correctly
• Not substituting the values of the binomial coefficients and the powers of $x$ and $54$ correctly
• Not simplifying the expression correctly

Q: How can I practice expanding expressions like $(x + 54)^5$?

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A: You can practice expanding expressions like $(x + 54)^5$ by:

• Using online resources and calculators to check your work
• Working with different values of $a$, $b$, and $n$
• Practicing with different types of expressions, such as $(x - 54)^5$ or $(x + 27)^3$

Conclusion

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In this article, we answered some common questions related to expanding the expression $(x + 54)^5$. We covered topics such as the binomial theorem, calculating binomial coefficients, and simplifying expressions. We also provided some tips and resources for practicing expanding expressions like $(x + 54)^5$.