The correct answer and explanation is:
Answer:
For this expression, a = 7, b = 2, and c = 9.
Explanation
The problem requires converting an expression with a fractional exponent into its equivalent radical form. The original expression given is 7 raised to the power of 2/9, which is written as 7^(2/9). The goal is to represent this in the form c√(a^b) and identify the corresponding values of a, b, and c.
The fundamental rule for converting between fractional exponents and radicals is:
x^(m/n) = n√(x^m)
In this formula:
- x is the base.
- m is the numerator of the fractional exponent, which becomes the power of the base inside the radical.
- n is the denominator of the fractional exponent, which becomes the index, or root, of the radical.
Let’s apply this rule to the given expression, 7^(2/9).
First, we identify the components of our expression in relation to the general formula:
- The base, x, is 7.
- The numerator of the exponent, m, is 2.
- The denominator of the exponent, n, is 9.
Now, we substitute these values into the radical form n√(x^m):
- The denominator, n = 9, becomes the index of the root.
- The base, x = 7, is placed inside the radical sign.
- The numerator, m = 2, becomes the exponent of the base inside the radical.
Following this substitution, the expression 7^(2/9) is rewritten as 9√(7^2).
Finally, we compare this result, 9√(7^2), to the target form provided in the problem, c√(a^b).
By matching the parts of the two expressions:
- a, the base inside the radical, corresponds to 7.
- b, the exponent inside the radical, corresponds to 2.
- c, the index of the root, corresponds to 9.
Therefore, the correct values are a = 7, b = 2, and c = 9.
