To simplify the expression [tex]\( 8 \sqrt{8} + 6 \sqrt{72} \)[/tex], follow these steps:
1. Simplify the square roots:
- [tex]\( \sqrt{8} \)[/tex] can be broken down as [tex]\( \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \)[/tex].
- [tex]\( \sqrt{72} \)[/tex] can be broken down as [tex]\( \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \)[/tex].
2. Substitute these simplified forms back into the expression:
- For [tex]\( 8 \sqrt{8} \)[/tex], substitute [tex]\( \sqrt{8} \)[/tex] with [tex]\( 2\sqrt{2} \)[/tex]:
[tex]\[
8 \sqrt{8} = 8 \cdot 2\sqrt{2} = 16\sqrt{2}.
\][/tex]
- For [tex]\( 6 \sqrt{72} \)[/tex], substitute [tex]\( \sqrt{72} \)[/tex] with [tex]\( 6\sqrt{2} \)[/tex]:
[tex]\[
6 \sqrt{72} = 6 \cdot 6\sqrt{2} = 36\sqrt{2}.
\][/tex]
3. Combine the terms:
- Add the two terms, [tex]\( 16\sqrt{2} \)[/tex] and [tex]\( 36\sqrt{2} \)[/tex]:
[tex]\[
16\sqrt{2} + 36\sqrt{2} = (16 + 36)\sqrt{2} = 52\sqrt{2}.
\][/tex]
4. Calculate the numerical value:
- The numerical value for [tex]\( 16\sqrt{2} \)[/tex] is approximately [tex]\( 22.627 \)[/tex].
- The numerical value for [tex]\( 36\sqrt{2} \)[/tex] is approximately [tex]\( 50.912 \)[/tex].
- Adding these, the numerical value of [tex]\( 52\sqrt{2} \)[/tex] is approximately [tex]\( 73.539 \)[/tex].
Thus, the simplified form of the expression [tex]\( 8 \sqrt{8} + 6 \sqrt{72} \)[/tex] is [tex]\( 52\sqrt{2} \)[/tex], and its approximate numerical value is [tex]\( 73.539 \)[/tex].
