Question

СРОЧНО! Порівняйте значення виразів: а) 126^12 і 24^18; б) 31^11 і 17^14; в) 48^25 і 344^17 . Відповідь обґрунтуйте.

Answer 1

Ответ:

Объяснение:

а) 126^12 и 24^18

126^12 = (6*3*7)^12 = 6^12*3^12*7^12 = 6^12*3^6*3^6*7^12

24^18 = 6^18*4^18 = 6^12*6^6*4^18 = 6^12*3^6*2^6*64^6

Одинаковые части 6^12*3^6 можно сократить. Остается сравнить:

3^6*7^12 и 2^6*64^6

3^6*7^12 = 3^6*49^6 = 147^6

2^6*64^6 = 128^6

Ясно, что 147^6 > 128^6, поэтому:

126^12 > 24^18

б) 31^11 и 17^14

31^11 < 32^11 = 16^11*2^11

17^14 > 16^14 = 16^11*16^3

Сократим 16^11 и сравним 2^11 и 16^3

16^3 = (2^4)^3 = 2^12 > 2^11

Получаем:

31^11 < 32^11 = 16^11*2^11 < 16^11*16^3 = 16^14 < 17^14

31^11 < 17^14

в) 48^25 и 344^17

48^25 = 48^17*48^8

344^17 = 8^17*43^17 > 8^17*42^17

8^17*42^17 = 8^17*6^17*7^17 = 48^17*7^17

Сократим 48^17 и сравним 48^8 и 7^17:

7^17 = 7*7^16 = 7*(7^2)^8 = 7*49^8 > 48^8

Получаем:

344^17 > 8^17*42^17 = 48^17*7^17 > 48^17*48^8 = 48^25

48^25 < 344^17

Answer 2

Comparing Exponential Expressions

When comparing the values of exponential expressions, it's important to consider the base and the exponent. Let's compare the given expressions:

а) 126^12 and 24^18: The value of 126^12 is significantly larger than 24^18. This is because the base 126 is larger than 24, and even though the exponent for 24 is larger, the base for 126^12 being significantly larger results in a larger value.

б) 31^11 and 17^14: In this case, 17^14 is larger than 31^11. The reason for this is that although the base 31 is larger than 17, the exponent 14 is significantly larger than 11, resulting in a larger value for 17^14.

в) 48^25 and 344^17: The value of 344^17 is larger than 48^25. This is because the base 344 is significantly larger than 48, and even though the exponent for 48 is larger, the base for 344^17 being significantly larger results in a larger value.

In summary, when comparing exponential expressions, the magnitude of the base and the exponent both play crucial roles in determining the relative size of the values.

I hope this helps! If you have further questions or need additional clarification, feel free to ask!