Which expression is equivalent to [tex]$\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}$[/tex] ?
A. [tex]$x^{\frac{2}{9}}$[/tex]
B. [tex]$x^{\frac{2}{3}}$[/tex]
C. [tex]$x^{\frac{8}{27}}$[/tex]
D. [tex]$x^{\frac{7}{3}}$[/tex]
Answer (1)
Simplify inside the parenthesis by adding the exponents: x 3 4 x 3 2 = x 3 6 = x 2 .
Substitute the simplified expression back into the original expression: ( x 2 ) 3 1 .
Simplify the power by multiplying the exponents: x 2 ⋅ 3 1 = x 3 2 .
The equivalent expression is x 3 2 .
Explanation
Understanding the problem We are asked to find an expression equivalent to ( x 3 4 x 3 2 ) 3 1 . To do this, we will use the properties of exponents to simplify the expression.
Simplifying inside the parenthesis First, we simplify the expression inside the parenthesis. When multiplying exponential terms with the same base, we add the exponents: x a x b = x a + b . In our case, we have x 3 4 x 3 2 = x 3 4 + 3 2 = x 3 6 = x 2 .
Substituting back Now we substitute the simplified expression back into the original expression: ( x 3 4 x 3 2 ) 3 1 = ( x 2 ) 3 1 .
Simplifying the power Next, we simplify the expression using the rule that when raising an exponential term to a power, we multiply the exponents: ( x a ) b = x ab . In our case, we have ( x 2 ) 3 1 = x 2 ⋅ 3 1 = x 3 2 .
Final Answer Therefore, the expression equivalent to ( x 3 4 x 3 2 ) 3 1 is x 3 2 .
Examples
Understanding exponential expressions is crucial in various fields, such as calculating growth rates in finance or modeling radioactive decay in physics. For instance, if a population of bacteria grows by a factor of x 3 2 every hour, this expression helps us predict the population size after a certain number of hours. By simplifying complex exponential expressions, we can make accurate predictions and informed decisions in real-world scenarios.
