Simplify.
[tex]49^{\frac{5}{2}}[/tex]
Answer (1)
Rewrite 49 as 7 2 .
Apply the power of a power rule: ( 7 2 ) 2 5 = 7 2 ⋅ 2 5 = 7 5 .
Calculate 7 5 = 16807 .
The simplified expression is 16807 .
Explanation
Understanding the problem We are asked to simplify the expression 4 9 2 5 . This means we need to evaluate this expression and write the result as an integer or a fraction.
Rewriting the base First, we can rewrite the base 49 as 7 2 . This gives us ( 7 2 ) 2 5 .
Applying the power of a power rule Next, we use the power of a power rule, which states that ( a m ) n = a m ⋅ n . Applying this rule, we get 7 2 ⋅ 2 5 = 7 5 .
Calculating the power Now, we need to calculate 7 5 . This means 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 . We can calculate this as follows: 7 2 = 49 7 3 = 49 ⋅ 7 = 343 7 4 = 343 ⋅ 7 = 2401 7 5 = 2401 ⋅ 7 = 16807
Final result Therefore, 4 9 2 5 = 16807 .
Examples
Exponents are used to calculate compound interest. For example, if you invest money in an account that pays interest, the amount of money you have after a certain number of years can be calculated using an exponent. Also, exponents are used in scientific notation to represent very large or very small numbers, such as the distance to stars or the size of atoms. Understanding exponents helps in managing finances and understanding the scale of the universe.
