anemone said:Prove that $8^{91}\gt 7^{92}$.
kaliprasad said:$8^{91} = 7^{91} * ( 1 + \frac{1}{7})^{91}$
$> 7^{91} * ( 1 + \frac{91}{7})$ using for n integer > 1 $(1+x)^n > 1 + nx$ as other terms are positive
$> 7^{91} * 14 > 2* 7^{92} > 7^{92}$
kaliprasad said:$8^{91} = 7^{91} * ( 1 + \frac{1}{7})^{91}$
$> 7^{91} * ( 1 + \frac{91}{7})$ using for n integer > 1 $(1+x)^n > 1 + nx$ as other terms are positive
$> 7^{91} * 14 > 2* 7^{92} > 7^{92}$
The purpose of comparing two powers of numbers is to determine which number has a greater value or magnitude. This can be helpful in solving mathematical equations or evaluating different quantities.
To compare two powers of numbers, you can either use a calculator or use mathematical rules such as the product rule and quotient rule. These rules involve multiplying or dividing the numbers being raised to a power and comparing the results.
Yes, two powers of different numbers can be compared. However, it is important to note that the two numbers must have the same base in order to be compared. If the bases are different, the powers cannot be directly compared.
Comparing two powers of numbers can help in various real-life situations, such as in finance, science, and engineering. It can also aid in simplifying mathematical expressions and solving equations by identifying the larger or smaller value.
One limitation of comparing powers of numbers is that it only gives a relative comparison between the two numbers and does not provide an exact numerical value. Additionally, it is not applicable for comparing irrational or imaginary numbers.