8√8 can be written in the form 8k. Find the value of k.
The Correct Answer and Explanation is:
To express ( 8\sqrt{8} ) in the form of ( 8k ), we start by simplifying the expression ( 8\sqrt{8} ).
Step 1: Simplify ( \sqrt{8} )
The square root of 8 can be rewritten using its prime factorization:
[
8 = 4 \times 2 = 2^3
]
Thus,
[
\sqrt{8} = \sqrt{2^3} = \sqrt{(2^2) \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}
]
Step 2: Substitute ( \sqrt{8} ) back into the expression
Now, substituting this back into our expression gives:
[
8\sqrt{8} = 8 \cdot (2\sqrt{2}) = 16\sqrt{2}
]
Step 3: Write ( 16\sqrt{2} ) in the form ( 8k )
To express ( 16\sqrt{2} ) in the form ( 8k ), we notice that:
[
16\sqrt{2} = 8 \cdot 2\sqrt{2}
]
This shows that ( k ) can be identified as ( 2\sqrt{2} ).
Conclusion: Finding the value of ( k )
Thus, we can conclude that:
[
k = 2\sqrt{2}
]
Final Answer
The value of ( k ) is ( 2\sqrt{2} ).
Explanation
In mathematical expressions, simplification is a key step for clarity and understanding. The expression ( 8\sqrt{8} ) initially may seem complex, but by breaking it down into its components, we can derive a more straightforward representation.
The importance of expressing numbers in different forms often aids in understanding relationships between numbers and can be particularly helpful in algebra and calculus. For instance, being able to express something like ( 8\sqrt{8} ) in the form of ( 8k ) emphasizes the multiple ( k ) that represents the scale of the quantity relative to 8.
Furthermore, recognizing that ( \sqrt{8} ) simplifies to ( 2\sqrt{2} ) showcases the utility of square roots and simplification techniques, making it easier to manipulate and understand the expression in various mathematical contexts, whether solving equations or applying in higher mathematics. Thus, the approach taken to simplify ( 8\sqrt{8} ) leads us directly to the value of ( k ), enriching our mathematical toolkit.