What is the equivalent expression of $x^{\frac{1}{3}} x^{\frac{1}{4}}$ ?
A. $\sqrt[2]{x^7}$
B. $\sqrt[12]{x^7}$
C. $\sqrt[7]{x^2}$
D. $\sqrt[12]{x}$
Answer (1)
Use the property x a x b = x a + b to combine the exponents: x 3 1 x 4 1 = x 3 1 + 4 1 .
Add the exponents 3 1 and 4 1 by finding a common denominator: 3 1 + 4 1 = 12 4 + 12 3 = 12 7 .
Rewrite the expression as x 12 7 .
Convert the fractional exponent to a radical: x 12 7 = 12 x 7 .
12 x 7
Explanation
Understanding the problem We are given the expression x 3 1 x 4 1 and asked to find an equivalent expression from the given options. The key here is to use the properties of exponents to simplify the expression and match it with one of the provided choices.
Applying the exponent rule To simplify the expression, we use the property that when multiplying exponential terms with the same base, we add the exponents: x a x b = x a + b . In our case, a = 3 1 and b = 4 1 . So we need to calculate 3 1 + 4 1 .
Adding the fractions To add the fractions 3 1 and 4 1 , we need to find a common denominator. The least common multiple of 3 and 4 is 12. Thus, we rewrite the fractions with the common denominator of 12: 3 1 = 12 4 and 4 1 = 12 3 . Now we can add them: 12 4 + 12 3 = 12 4 + 3 = 12 7 .
Converting to radical form Now we can rewrite the original expression as x 12 7 . To match this with the given options, we need to remember that x n m is equivalent to n x m . Therefore, x 12 7 is equivalent to 12 x 7 .
Final Answer Comparing our simplified expression 12 x 7 with the given options, we find that it matches option B. Therefore, the equivalent expression of x 3 1 x 4 1 is 12 x 7 .
Examples
Understanding and simplifying expressions with fractional exponents is crucial in various fields, such as physics and engineering, where complex equations often involve roots and powers. For instance, when analyzing the behavior of waves or calculating the energy levels of quantum particles, you might encounter expressions with fractional exponents. Simplifying these expressions allows for easier manipulation and interpretation of the results, leading to a better understanding of the underlying phenomena. This skill is also useful in computer graphics when dealing with scaling and transformations.
