Question
13. Обчисліть * \( \frac{9^{8} \cdot 4^{5}}{27^{4} \cdot 16^{2}}= \)
Ask by Simmons Pollard. in Germany
Feb 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Результат: 324
Solution
Calculate or simplify the expression \( (9^8 * 4^5) / (27^4 * 16^2) \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{\left(9^{8}\times 4^{5}\right)}{\left(27^{4}\times 16^{2}\right)}\)
- step1: Remove the parentheses:
\(\frac{9^{8}\times 4^{5}}{27^{4}\times 16^{2}}\)
- step2: Factor the expression:
\(\frac{3^{16}\times 4^{5}}{3^{12}\times 16^{2}}\)
- step3: Reduce the fraction:
\(\frac{3^{4}\times 4^{5}}{16^{2}}\)
- step4: Factor the expression:
\(\frac{3^{4}\times 4^{5}}{4^{4}}\)
- step5: Reduce the fraction:
\(3^{4}\times 4\)
- step6: Evaluate the power:
\(81\times 4\)
- step7: Multiply:
\(324\)
Результат: \( \frac{9^{8} \cdot 4^{5}}{27^{4} \cdot 16^{2}} = 324 \).
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Давайте спростимо вираз. Перш ніж обчислити, перетворимо всі числа в степеневу форму: \( 9 = 3^{2},\) отже \( 9^{8} = (3^{2})^{8} = 3^{16},\) \( 4 = 2^{2},\) отже \( 4^{5} = (2^{2})^{5} = 2^{10},\) \( 27 = 3^{3},\) отже \( 27^{4} = (3^{3})^{4} = 3^{12},\) \( 16 = 2^{4},\) отже \( 16^{2} = (2^{4})^{2} = 2^{8}.\) Підставляючи це у вираз, отримаємо: \[ \frac{3^{16} \cdot 2^{10}}{3^{12} \cdot 2^{8}} = 3^{16 - 12} \cdot 2^{10 - 8} = 3^{4} \cdot 2^{2} = 81 \cdot 4 = 324. \] Отже, відповідь: \( \frac{9^{8} \cdot 4^{5}}{27^{4} \cdot 16^{2}} = 324. \)
