Simplify The Expression: 108 X 8 + 48 X 8 \sqrt{108 X^8} + \sqrt{48 X^8} 108 X 8 ​ + 48 X 8 ​ Enter The Exact Answer.Hints:- You Can Write Exponents With The ∧ \wedge ∧ Symbol, For Example, Writing X 8 X^8 X 8 As X ∧ 8 X^{\wedge}8 X ∧ 8 .- You Can Write Square Roots As

Simplify the Expression: $\sqrt{108 x^8} + \sqrt{48 x^8}$

Understanding the Problem

The given expression involves the addition of two square roots, each containing a variable $x$ raised to the power of 8. The first step is to simplify the expression by factoring out the common terms from the square roots.

Breaking Down the Square Roots

To simplify the expression, we need to break down the square roots into their prime factors. We can start by factoring the numbers inside the square roots.

$\sqrt{108 x^8} = \sqrt{2^2 \cdot 3^3 \cdot x^8}$

$\sqrt{48 x^8} = \sqrt{2^4 \cdot 3 \cdot x^8}$

Simplifying the Square Roots

Now that we have factored the numbers inside the square roots, we can simplify them by taking out the perfect squares.

$\sqrt{2^2 \cdot 3^3 \cdot x^8} = \sqrt{(2^2 \cdot x^4)^2 \cdot 3^3} = 2x^4 \sqrt{3^3}$

$\sqrt{2^4 \cdot 3 \cdot x^8} = \sqrt{(2^4 \cdot x^4)^2 \cdot 3} = 4x^4 \sqrt{3}$

Combining the Simplified Square Roots

Now that we have simplified the individual square roots, we can combine them by adding the two expressions.

$2x^4 \sqrt{3^3} + 4x^4 \sqrt{3}$

Factoring Out the Common Term

We can simplify the expression further by factoring out the common term $x^4$.

$x^4 (2 \sqrt{3^3} + 4 \sqrt{3})$

Simplifying the Expression

Now that we have factored out the common term, we can simplify the expression by evaluating the square roots.

$x^4 (2 \sqrt{27} + 4 \sqrt{3})$

Evaluating the Square Roots

We can simplify the expression further by evaluating the square roots.

$x^4 (2 \cdot 3 \sqrt{3} + 4 \sqrt{3})$

Combining Like Terms

We can simplify the expression further by combining like terms.

$x^4 (6 \sqrt{3} + 4 \sqrt{3})$

Final Simplification

Now that we have combined like terms, we can simplify the expression further.

$x^4 (10 \sqrt{3})$

The Final Answer

The final answer is $\boxed{x^4 (10 \sqrt{3})}$.

Conclusion

In this article, we simplified the expression $\sqrt{108 x^8} + \sqrt{48 x^8}$ by breaking down the square roots into their prime factors, simplifying them by taking out the perfect squares, combining the simplified square roots, factoring out the common term, and finally simplifying the expression by evaluating the square roots and combining like terms. The final answer is $x^4 (10 \sqrt{3})$.
Simplify the Expression: $\sqrt{108 x^8} + \sqrt{48 x^8}$ - Q&A

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the simplification of the expression $\sqrt{108 x^8} + \sqrt{48 x^8}$.

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to break down the square roots into their prime factors.

Q: How do I factor the numbers inside the square roots?

A: To factor the numbers inside the square roots, you need to find the prime factors of the numbers. For example, the prime factors of 108 are 2^2 and 3^3, and the prime factors of 48 are 2^4 and 3.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2. A non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I simplify the square roots by taking out the perfect squares?

A: To simplify the square roots by taking out the perfect squares, you need to identify the perfect squares inside the square roots and take them out. For example, in the expression $\sqrt{108 x^8}$, the perfect square is $2^2 \cdot x^4$, which can be taken out as $2x^4$.

Q: What is the difference between combining like terms and combining unlike terms?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Combining unlike terms involves adding or subtracting terms that have different variables or exponents.

Q: How do I combine like terms in the expression?

A: To combine like terms in the expression, you need to identify the like terms and add or subtract them. For example, in the expression $x^4 (6 \sqrt{3} + 4 \sqrt{3})$, the like terms are $6 \sqrt{3}$ and $4 \sqrt{3}$, which can be combined as $10 \sqrt{3}$.

Q: What is the final answer to the expression?

A: The final answer to the expression is $x^4 (10 \sqrt{3})$.

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it helps to make the expression easier to understand and work with. It also helps to identify any patterns or relationships between the variables and constants in the expression.

Q: How can I apply the concepts learned in this article to other problems?

A: The concepts learned in this article can be applied to other problems involving the simplification of expressions. You can use the same steps and techniques to simplify other expressions, such as $\sqrt{a^2 + b^2}$ or $\sqrt{a^2 - b^2}$.

Conclusion

In this article, we answered some of the most frequently asked questions related to the simplification of the expression $\sqrt{108 x^8} + \sqrt{48 x^8}$. We covered topics such as factoring, perfect squares, combining like terms, and the importance of simplifying expressions. We hope that this article has been helpful in understanding the concepts and techniques involved in simplifying expressions.