Which expression is equivalent to $\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}} ?$
A. $x^{\frac{2}{9}}$
B. $x^{\frac{2}{3}}$
C. $x^{\frac{8}{27}}$
D. $x^{\frac{7}{3}}$
Answer (1)
First, combine the terms inside the parenthesis by adding the exponents: x 3 4 x 3 2 = x 3 6 = x 2 .
Next, apply the outer exponent by multiplying the exponents: ( x 2 ) 3 1 = x 2 × 3 1 = x 3 2 .
The simplified expression is x 3 2 .
Therefore, 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 properties of exponents to simplify the expression.
Simplifying Inside 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 . Therefore, x 3 4 x 3 2 = x 3 4 + 3 2 = x 3 6 = x 2 .
Simplifying the Outer Exponent Now we have ( x 2 ) 3 1 . When raising an exponential term to a power, we multiply the exponents: ( x a ) b = x ab . Therefore, ( x 2 ) 3 1 = x 2 × 3 1 = x 3 2 .
Final Answer Thus, the expression ( x 3 4 x 3 2 ) 3 1 simplifies to x 3 2 .
Examples
Understanding and simplifying exponential expressions is crucial in various fields, such as physics and engineering. For example, in physics, the energy E of a photon is related to its frequency f by the equation E = h f , where h is Planck's constant. If we express the frequency as a function of wavelength λ using f = c / λ (where c is the speed of light), we can rewrite the energy equation as E = h c / λ = h c λ − 1 . Simplifying such expressions using exponent rules helps in analyzing the relationships between different physical quantities.
