To simplify the expression [tex]\( 4 \sqrt{75} + \sqrt{48} \)[/tex], let's break it down into steps:
1. Simplifying [tex]\( \sqrt{75} \)[/tex]:
- Notice that [tex]\( 75 \)[/tex] can be factored into [tex]\( 25 \times 3 \)[/tex].
- Therefore, [tex]\( \sqrt{75} = \sqrt{25 \times 3} \)[/tex].
- This can be further simplified to [tex]\( \sqrt{25} \cdot \sqrt{3} \)[/tex].
- Since [tex]\( \sqrt{25} = 5 \)[/tex], we have [tex]\( \sqrt{75} = 5 \sqrt{3} \)[/tex].
2. Simplifying [tex]\( \sqrt{48} \)[/tex]:
- Notice that [tex]\( 48 \)[/tex] can be factored into [tex]\( 16 \times 3 \)[/tex].
- Therefore, [tex]\( \sqrt{48} = \sqrt{16 \times 3} \)[/tex].
- This can be further simplified to [tex]\( \sqrt{16} \cdot \sqrt{3} \)[/tex].
- Since [tex]\( \sqrt{16} = 4 \)[/tex], we have [tex]\( \sqrt{48} = 4 \sqrt{3} \)[/tex].
3. Combining the simplified terms:
- Substitute back the simplified forms into the original expression.
- This gives us [tex]\( 4 \sqrt{75} + \sqrt{48} = 4 (5 \sqrt{3}) + 4 \sqrt{3} \)[/tex].
4. Distribute the coefficients and combine like terms:
- [tex]\( 4 (5 \sqrt{3}) = 20 \sqrt{3} \)[/tex].
- Thus, the expression becomes [tex]\( 20 \sqrt{3} + 4 \sqrt{3} \)[/tex].
5. Combining the coefficients of [tex]\( \sqrt{3} \)[/tex]:
- Add the coefficients: [tex]\( 20 \sqrt{3} + 4 \sqrt{3} = (20 + 4) \sqrt{3} = 24 \sqrt{3} \)[/tex].
6. Numerical values (given results):
- [tex]\( \sqrt{75} \approx 5.000 \)[/tex]
- [tex]\( \sqrt{48} \approx 4.000 \)[/tex]
Finally, substituting the values back, we get the numerical simplified result of approximately 41.569.
So, the simplified form of the expression [tex]\( 4 \sqrt{75} + \sqrt{48} \)[/tex] in radical terms is [tex]\( 24 \sqrt{3} \)[/tex], and numerically it approximates to 41.569.
