Question

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{p^{2}}{q^{0}}\right)^{-1}$$

Short answer

\frac{1}{p^{2}}

Step-by-step solution

Simplify the Base

First, simplify the base of the expression by evaluating any exponents and applying the properties of exponents. Since any number raised to the zero power is 1, we have that \( q^{0} = 1 \). This simplifies the base to \( \frac{p^{2}}{1} \), which is simply \( p^{2} \).

Apply the Negative Exponent Rule

Next, we apply the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^{n}} \), to the entire expression. Thus \( \left(\frac{p^{2}}{q^{0}}\right)^{-1} = \frac{1}{\left(\frac{p^{2}}{q^{0}}\right)^{1}} \), which simplifies down to \( \frac{1}{p^{2}} \).

Write Without Negative Exponents

The expression \( \frac{1}{p^{2}} \) already has no negative exponents, so this is the simplified form of the original problem without negative exponents.

Key concepts

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying mathematical expressions. Exponents, which are often called powers, indicate how many times a number, known as the base, is multiplied by itself. There are several key properties that help in solving exponential expressions:

• Product Rule: To multiply two exponents with the same base, you add the exponents. For instance, \( a^m \times a^n = a^{m+n} \).
• Quotient Rule: To divide two exponents with the same base, you subtract the exponents. For example, \( a^m \div a^n = a^{m-n} \).
• Power of a Power Rule: To raise an exponent to another power, you multiply the exponents. This is expressed as \( (a^m)^n = a^{m \times n} \).
• Zero Exponent Rule: Any base raised to the zero power is equal to one, summarized as \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent, as in \( a^{-n} = \frac{1}{a^n} \).

These properties can be utilized in tandem to simplify complex expressions, making them manageable and preparing them for further mathematical operations like solving equations or applying functions.

Negative Exponent Rule

The negative exponent rule is a pivotal concept in managing expressions with exponents. By definition, a negative exponent represents the reciprocal of the base raised to the opposite positive exponent. For example, \( x^{-n} = \frac{1}{x^n} \). When simplifying an expression with a negative exponent, you effectively move the base to the opposite side of the fraction line and change the sign of the exponent to positive.

For instance, with the base expression simplified to \( p^2 \), applying the negative exponent rule to \( (p^2)^{-1} \) transforms it to \( \frac{1}{(p^2)^1} \) or simply \( \frac{1}{p^2} \), which is a positive exponent form. It's important to apply this rule correctly because it directly influences how expressions are simplified and interpreted in further calculations.

Solving Exponential Equations

When faced with exponential equations, the goal is often to solve for the variable in the exponent. This process usually involves using the properties of exponents to rewrite the equation in a form that facilitates the isolation of the variable. Strategies for solving exponential equations include equating bases and using logarithms.

For example, if you had the equation \( a^x = a^y \), based on the property that if the bases are the same, the exponents must be equal, you could deduce that \( x = y \). If the bases are not the same, you could potentially use logarithms to solve for the variable.

Understanding how to manipulate expressions using the properties of exponents becomes especially handy here. This knowledge allows one to simplify and compare components of the equation effectively, which is a vital step in finding the solution.