Rewrite without rational exponents, and simplify, if possible.
$81^{\frac{3}{4}}$
Answer (1)
Rewrite 81 as 3 4 .
Substitute 81 = 3 4 into the expression to get ( 3 4 ) 4 3 .
Use the power of a power rule: ( a m ) n = a mn to simplify the expression to 3 4 ⋅ 4 3 = 3 3 .
Calculate 3 3 = 27 , so the final answer is 27 .
Explanation
Understanding the Problem We are given the expression 8 1 4 3 . Our goal is to rewrite this expression without rational exponents and simplify it.
Rewriting 81 as a Power of 3 First, we can rewrite 81 as a power of 3. Since 3 4 = 81 , we can substitute this into the expression: 8 1 4 3 = ( 3 4 ) 4 3
Applying the Power of a Power Rule Now, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we get: ( 3 4 ) 4 3 = 3 4 ⋅ 4 3
Simplifying the Exponent Next, we simplify the exponent: 4 ⋅ 4 3 = 3 So the expression becomes: 3 3
Calculating the Final Value Finally, we calculate 3 3 : 3 3 = 3 ⋅ 3 ⋅ 3 = 27 Therefore, 8 1 4 3 = 27 .
Examples
Understanding rational exponents is useful in various fields, such as calculating growth rates or dealing with scaling factors in geometry. For instance, if you are analyzing the growth of a population that increases by a factor of 8 1 4 3 over a certain period, you can simplify this to find that the population actually increases by a factor of 27. This simplification makes it easier to understand and communicate the actual growth rate.
