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In mathematical expressions, when we have a fraction raised to a power, we deal with both the numerator and the denominator separately. For instance, if we encounter an expression like \(\bigg(\frac{a}{b}\bigg)^n\), we apply the exponent \(n\) to both \(a\) and \(b\): \(\biggah\frac{a^n}{b^n}\bigg\). Let's examine this with an example: Suppose we have \(\big(\frac{3}{4}\big)^2\). We would compute \(3^2\) and \(4^2\) separately, leading to \(\frac{9}{16}\). This separation helps to simplify complex fraction exponents.

Simplifying expressions involves breaking down complex mathematical phrases into simpler forms. When dealing with fractions and exponents, follow these steps to simplify:

• Identify the base and the exponent.
• Apply the exponent to both parts of the fraction separately.
• Calculate the powers for both the numerator and the denominator.
• Combine the results to form the final expression.

For example, let's consider the expression \(\big(-\frac{3}{4}\big)^3\).

• Identify \(-\frac{3}{4}\) as the base and \(3\) as the exponent.
• Apply the exponent to \(-\frac{3}{4}\): \((-3)^3\) and \(4^3\).
• Calculate separately: \((-3)^3 = -27\) and \(4^3 = 64\).
• Combine them to get \(\frac{-27}{64}\).

Following this method ensures you understand each step, making the process more manageable.

When working with negative base powers, it's important to remember how exponents affect the sign and value of the base. If you have a negative base raised to an odd exponent, the result will be negative. Conversely, raising a negative base to an even exponent results in a positive number. Consider the example \((-2)^3\): \((-2)^3\) means multiplying -2 by itself three times: \(-2 \times -2 \times -2 = -8\). Notice the negative sign remains because three is odd. However, if the exponent were even: \((-2)^4\), it would be \(-2 \times -2 \times -2 \times -2 = 16\), a positive result as the negatives cancel out in pairs. Thus, keeping track of odd and even exponents helps in understanding the sign of the final value when working with negative bases.