Solve for $m$. $m^{\frac{3}{4}}=27$
Answer (2)
Raise both sides of the equation to the power of 3 4 : m = 2 7 3 4 .
Rewrite 27 as 3 3 : m = ( 3 3 ) 3 4 .
Simplify the exponent: m = 3 3 ⋅ 3 4 = 3 4 .
Calculate 3 4 to find the value of m : m = 81 . The final answer is 81 .
Explanation
Understanding the Problem We are given the equation m 4 3 = 27 and asked to find the value of m . To do this, we need to isolate m on one side of the equation.
Isolating m To isolate m , we raise both sides of the equation to the power of 3 4 . This is the reciprocal of the exponent of m , and will cancel out the exponent on m . So we have: ( m 4 3 ) 3 4 = 2 7 3 4 m = 2 7 3 4
Simplifying the Exponent Now we need to evaluate 2 7 3 4 . We can rewrite 27 as 3 3 , so we have: m = ( 3 3 ) 3 4 Using the power of a power rule, we multiply the exponents: m = 3 3 ⋅ 3 4 = 3 4
Calculating the Value of m Finally, we calculate 3 4 :
m = 3 4 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 9 ⋅ 9 = 81 So, the value of m is 81.
Examples
Understanding fractional exponents is useful in various fields, such as calculating growth rates or decay rates in biology or finance. For example, if a population grows by a factor of x 2 1 each year, after 5 years, the total growth factor would be ( x 2 1 ) 5 = x 2 5 . This concept is also used in calculating compound interest or depreciation of assets.
To solve for m in the equation m 4 3 = 27 , we raise both sides to the power of 3 4 , simplify 27 as 3 3 , and ultimately find that m = 81 .
;
