Which expression is equivalent to (x3/7) ²
A. x 6/14
B. x 17/7
C. x 6/7
D. x 5/7
The Correct answer and Explanation is:
To solve the expression ((x^{3/7})^2), we use the exponentiation rule ((a^m)^n = a^{m \cdot n}). This rule states that when you raise a power to another power, you multiply the exponents.
Step-by-step solution:
- The original expression is ((x^{3/7})^2).
- Using the rule ((a^m)^n = a^{m \cdot n}), we multiply the exponents:
[
x^{3/7 \cdot 2} = x^{6/7}
]
So, the expression simplifies to (x^{6/7}).
Now, let’s analyze the answer choices:
A. (x^{6/14}) — This expression simplifies to (x^{3/7}), but it does not match our result of (x^{6/7}).
B. (x^{17/7}) — The exponent is incorrect and does not match the simplified form (x^{6/7}).
C. (x^{6/7}) — This is exactly what we calculated as the simplified form of the expression ((x^{3/7})^2).
D. (x^{5/7}) — The exponent is incorrect, as it does not match the calculated (x^{6/7}).
Correct Answer:
The correct answer is C. (x^{6/7}).
Explanation of the exponentiation rule:
The exponentiation rule ((a^m)^n = a^{m \cdot n}) is a fundamental property of exponents. It helps simplify expressions where one exponent is raised to another power. In this case, (x^{3/7}) was raised to the power of 2, which means multiplying the exponents: (\frac{3}{7} \cdot 2 = \frac{6}{7}).
This simplification process can be applied to other expressions with fractional exponents as well. By multiplying the fractional exponents, you efficiently reduce complex powers into simpler forms that are easier to work with or compare to given choices.
In this particular problem, the answer (x^{6/7}) demonstrates how exponents operate under multiplication and how fractional exponents follow the same rules as whole-number exponents.