Simplify -4 * sqrt(192x)
Understand the Problem
The question asks to simplify the expression -4 * sqrt(192x). This involves factoring 192 and taking out any perfect squares from under the square root symbol. This is a math question from Algebra.
Answer
$-32\sqrt{3x}$
Answer for screen readers
$-32\sqrt{3x}$
Steps to Solve
- Factor 192
Find the prime factorization of 192: $192 = 2^6 \cdot 3$
- Rewrite the expression using the factorization
Replace 192 with its prime factorization within the square root: $-4\sqrt{192x} = -4\sqrt{2^6 \cdot 3 \cdot x}$
- Simplify the square root
Simplify the square root by taking out perfect squares. Since $2^6$ is a perfect square $(2^3)^2 = 2^6 = 64$, we can take $2^3 = 8$ out of the square root: $-4\sqrt{2^6 \cdot 3 \cdot x} = -4 \cdot 2^3 \sqrt{3x} = -4 \cdot 8 \sqrt{3x}$
- Multiply the constants
Multiply the constants outside the square root: $-4 \cdot 8 \sqrt{3x} = -32\sqrt{3x}$
$-32\sqrt{3x}$
More Information
The simplified form of $-4\sqrt{192x}$ is $-32\sqrt{3x}$.
Tips
A common mistake is not fully factoring the number under the square root, which leads to not extracting the largest possible perfect square. For example, someone might stop at $\sqrt{192} = \sqrt{4 \cdot 48}$ and only take out a 2, instead of fully factoring to find $\sqrt{192} = \sqrt{64 \cdot 3}$ and taking out an 8.
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