The Correct Answer and Explanation is:
The correct answer is 5√5.
To write the expression 5^(3/2) in surd form, we need to understand the relationship between fractional exponents and radicals. The general rule for converting a fractional exponent to a surd is x^(a/b) = ᵇ√(xᵃ). In this rule, ‘x’ is the base, ‘a’ is the numerator of the exponent, and ‘b’ is the denominator of the exponent. The denominator ‘b’ becomes the index of the root, and the numerator ‘a’ becomes the power of the base inside the root.
Let’s apply this rule to the given expression, 5^(3/2):
Here, the base (x) is 5.
The numerator of the exponent (a) is 3.
The denominator of the exponent (b) is 2.
Substituting these values into the formula, we get:
5^(3/2) = ²√(5³)
A root with an index of 2 is a square root, which is typically written without the ‘2’ in front of the radical symbol. So, the expression is:
√(5³)
Next, we calculate the value inside the radical. 5³ is equal to 5 × 5 × 5, which is 125.
So, our expression becomes:
√125
This is a correct surd, but it is not in its simplest form. To simplify a surd, we must find the largest perfect square that is a factor of the number inside the root (the radicand). For 125, the factors are 1, 5, 25, and 125. The largest perfect square factor is 25, because 5² = 25.
We can rewrite √125 as the square root of the product of its factors:
√125 = √(25 × 5)
Using the multiplication property of radicals, which states that √(m × n) = √m × √n, we can split this into two separate roots:
√(25 × 5) = √25 × √5
Since √25 is equal to 5, we can simplify the expression to:
5 × √5
Therefore, the final simplified surd form of 5^(3/2) is 5√5.
