Simplify fractional indices, step by step explanation.

To simplify fractional indices, follow these steps:

Step 1: Rewrite the fractional index as a root. For example, if you have the expression x^(2/3), rewrite it as the cube root of x squared (the numerator is the power and the denominator is the root).

Step 2: Simplify the root. In the example above, simplify the cube root of x squared by taking the cube root of x and squaring it.

Step 3: If possible, simplify further by combining like terms or simplifying any remaining radicals.

For example, if you have the expression (x^2 * y^(1/2))^(1/3), follow these steps:

Step 1: Rewrite the fractional index as a root: (x^2 * √y)^(1/3)

Step 2: Simplify the root: cube root of x squared times the cube root of the square root of y

Step 3: Combine like terms if possible: cube root of x squared times the square root of y

This is the simplified form of the expression.

Step 1: Convert the number with the fractional index to radical form.

For example, to simplify \( x^{1/2} \), convert it to \( \sqrt{x} \).

Step 2: Rewrite the expression with a rational exponent.

The rational exponent is the denominator of the fractional exponent.

For example, \( \sqrt{x} = x^{1/2} \) has a rational exponent of \( 1/2 \).

Step 3: Apply the laws of exponents.

- Product rule: \( x^a \cdot x^b = x^{a+b} \)
- Power rule: \( (x^a)^b = x^{a\cdot b} \)

Step 4: Simplify the expression by combining like terms.

For example, to simplify \( x^{1/2} \cdot x^{1/4} \), use the product rule to get \( x^{1/2 + 1/4} \). Then, simplify \( 1/2 + 1/4 = 3/4 \) to get \( x^{3/4} \).

Additional Examples:

- \( x^{-1/2} = \frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}} \)
- \( (x^2)^{1/3} = x^{2/3} \)
- \( \frac{x^{1/4} \cdot x^{3/4}}{x^{1/2}} = \frac{x^{4/4}}{x^{1/2}} = \frac{x}{x^{1/2}} = x^{1/2} \)

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