Problem 14 - Elementary and Intermediate Algebra

Question

Complete each rule for exponents. $$ \frac{1}{x^{-m / n}}= \square $$

Step-by-Step Solution

Answer

\(\frac{1}{x^{-m/n}} = x^{m/n}\).

Step 1: Recall the Negative Exponent Rule

The negative exponent rule states that a negative exponent indicates a reciprocal. Specifically, for any non-zero number \(a\) and any integer \(n\), \(a^{-n} = \frac{1}{a^n}\). In our problem, the exponent is \(-\frac{m}{n}\).

Step 2: Convert the Negative Exponent

Apply the negative exponent rule to \(x^{-m/n}\). According to the rule, \(x^{-m/n} = \frac{1}{x^{m/n}}\).

Step 3: Find the Reciprocal

Since \( \frac{1}{x^{-m/n}} \) is the expression we started with, we need to find the reciprocal of \(x^{-m/n}\). Based on Step 2, the reciprocal of \(x^{-m/n}\) is \(x^{m/n}\).

Step 4: Verify the Solution

The expression \(\frac{1}{x^{-m/n}} = x^{m/n}\) can be verified by substituting back and ensuring both sides are equivalent. This confirms that the rule for the negative exponent is correctly applied.

Key Concepts

Negative Exponent Rule

The negative exponent rule is a fundamental concept in the world of exponents. When you see an exponent that is negative, it's a signal to take the reciprocal of the base. This means you "flip" the base from the numerator to the denominator or vice versa. For example, any non-zero number \(a\) with a negative exponent \(-n\) can be expressed as \(a^{-n} = \frac{1}{a^n}\).
To understand why this works, imagine that negative exponents are like reminders to undo multiplication. So, when you have a number multiplied by itself "negative" times, you actually need to divide.

• Example: \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
• As another illustration: \(x^{-1} = \frac{1}{x}\)

Whenever faced with a negative exponent, remember: take the reciprocal and convert the negative exponent to a positive one by changing the position in a fraction.

Reciprocal

A reciprocal is what you get when you flip a fraction. If the original number is \(x\), its reciprocal is \(\frac{1}{x}\). This concept is tightly woven into the fabric of exponent rules, especially when dealing with negatives. When you encounter a negative exponent, you essentially need the reciprocal of the base.
This flipping mechanism is like a two-way street. If you begin with \(\frac{1}{x}\), the reciprocal is \(x\).

• For instance, the reciprocal of \(5\) is \(\frac{1}{5}\).
• If you start with \(\frac{1}{4}\), its reciprocal is \(4\).

The idea of reciprocal is crucial because it simplifies the process of dealing with negative exponents. Applying the negative exponent rule involves recognizing the need for a reciprocal based switch.

Rational Exponents

Rational exponents are another interesting concept to unravel. When the exponent is a fraction, like \(\frac{m}{n}\), it indicates both rooting and raising to a power simultaneously. The number \(m\) in the fraction \(\frac{m}{n}\) serves to raise the base to a power, while \(n\) denotes the root.
This means \(x^{\frac{m}{n}}\) is equivalent to taking the \(n\)-th root of \(x\) and then raising it to the \(m\)-th power. Alternatively, it can mean raising \(x\) to the \(m\)-th power first and then finding the \(n\)-th root.

• For example: \(x^{\frac{1}{2}}\) is the square root of \(x\), and \(x^{\frac{3}{2}}\) can be seen as \(\sqrt{x^3}\) or \((\sqrt{x})^3\).
• Another way: \(x^{\frac{2}{3}}\) can be \((\sqrt[3]{x})^2\) or \(\sqrt[3]{x^2}\).

Understanding rational exponents is crucial when simplifying expressions and solving equations involving fractional powers.