To find the difference [tex]\( 6 x \sqrt{14 x} - 5 \sqrt{56 x^3} \)[/tex] in simplest form, let's break it down step by step.
1. Simplify each part of the expression:
- For [tex]\( 6 x \sqrt{14 x} \)[/tex]:
[tex]\[
6 x \sqrt{14 x}
\][/tex]
Recognize that [tex]\( \sqrt{14 x} = \sqrt{14} \cdot \sqrt{x} \)[/tex], so:
[tex]\[
6 x \sqrt{14 x} = 6 x \cdot \sqrt{14} \cdot \sqrt{x} = 6 \sqrt{14} \cdot x \cdot x^{1/2} = 6 \sqrt{14} \cdot x^{3/2}
\][/tex]
Thus, the simplified form of [tex]\( 6 x \sqrt{14 x} \)[/tex] is:
[tex]\[
6 \sqrt{14} \cdot x^{3/2}
\][/tex]
- For [tex]\( 5 \sqrt{56 x^3} \)[/tex]:
[tex]\[
5 \sqrt{56 x^3}
\][/tex]
Recognize that [tex]\( 56 x^3 = 56 \cdot x^3 \)[/tex] and [tex]\( \sqrt{56 x^3} = \sqrt{56} \cdot \sqrt{x^3} \)[/tex]. Also, [tex]\( \sqrt{x^3} = x^{3/2} \)[/tex], so:
[tex]\[
5 \sqrt{56 x^3} = 5 \sqrt{56} \cdot x^{3/2}
\][/tex]
Now, [tex]\( 56 = 4 \cdot 14 \)[/tex] and [tex]\( \sqrt{56} = \sqrt{4 \cdot 14} = \sqrt{4} \cdot \sqrt{14} = 2 \sqrt{14} \)[/tex], thus:
[tex]\[
5 \sqrt{56} \cdot x^{3/2} = 5 \cdot 2 \sqrt{14} \cdot x^{3/2} = 10 \sqrt{14} \cdot x^{3/2}
\][/tex]
So, the simplified form of [tex]\( 5 \sqrt{56 x^3} \)[/tex] is:
[tex]\[
10 \sqrt{14} \cdot x^{3/2}
\][/tex]
2. Find the difference of the simplified expressions:
[tex]\[
6 \sqrt{14} \cdot x^{3/2} - 10 \sqrt{14} \cdot x^{3/2}
\][/tex]
3. Combine like terms:
Both terms have the common factor [tex]\( \sqrt{14} \cdot x^{3/2} \)[/tex]:
[tex]\[
(6 \sqrt{14} \cdot x^{3/2}) - (10 \sqrt{14} \cdot x^{3/2})
\][/tex]
Factor out [tex]\( \sqrt{14} \cdot x^{3/2} \)[/tex]:
[tex]\[
\left(6 - 10\right) \sqrt{14} \cdot x^{3/2} = -4 \sqrt{14} \cdot x^{3/2}
\][/tex]
Therefore, the difference of [tex]\( 6 x \sqrt{14 x} - 5 \sqrt{56 x^3} \)[/tex] in simplest form is:
[tex]\[
-4 \sqrt{14} \cdot x^{3/2}
\][/tex]
