Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ p^{1-(a+b)} $$

Short answer

Question: Rewrite the expression \(p^{1-(a+b)}\) as a product or a quotient. Answer: \(\frac{p}{p^a \cdot p^b}\)

Step-by-step solution

Identify the exponent subtraction rule

We have an expression with an exponent rule that can be applied: $$ p^{1-(a+b)} $$ The exponent subtraction rule states that if \(n\) and \(m\) are integers, then: $$ a^{n-m} = \frac{a^n}{a^m} $$ In our expression, let \(n = 1\) and \(m = (a + b)\). Now, apply the exponent subtraction rule.

Apply the exponent subtraction rule

Applying the exponent subtraction rule to our expression, we get: $$ p^{1 - (a + b)} = \frac{p^1}{p^{a+b}} $$

Express the exponent in the denominator

Now, express the exponent in the denominator as a product: $$ \frac{p^1}{p^{a+b}} = \frac{p}{p^a \cdot p^b} $$ So, the expression \(p^{1 - (a+b)}\) can be written as a quotient: $$ p^{1-(a+b)} = \frac{p}{p^a \cdot p^b} $$

Key concepts

Product of Powers

In mathematics, the product of powers rule is a fundamental concept when dealing with exponents. This rule helps simplify expressions involving exponential terms with the same base. If you have two powers with the same base, you can simply add their exponents together. For example, if you have \( a^m \cdot a^n \), the product of powers rule states:\[a^m \cdot a^n = a^{m+n}\] Let's break it down:

• "Base" refers to the number or variable that is being raised to a power.
• The numbers \(m\) and \(n\) are the exponents indicating the power to which the base is raised.
• All that is required when using the product of powers is addition of the exponents.

The product of powers rule can simplify calculations, especially with large numbers or complex algebraic expressions. In our expression, however, it comes into play when we rewrite the denominator in terms of individual powers (i.e., \(p^a \cdot p^b\)). This allows us to clearly separate and understand the components.

Quotient of Powers

The quotient of powers rule is another essential exponent rule. This rule allows us to divide exponential expressions with the same base by subtracting the exponent of the divisor from the exponent of the dividend. In symbols, if \(a\) is a non-zero number and \(m\) and \(n\) are integers, the rule is presented as:\[\frac{a^m}{a^n} = a^{m-n}\]Here's a simplified explanation:

• The "dividend" is the number or expression being divided. In this case, it's \(a^m\).
• The "divisor" is the number or expression that the dividend is divided by, here it's \(a^n\).
• Subtract the exponent of the divisor from the exponent of the dividend.
• The base remains unchanged.

In the original exercise, the expression \(p^{1-(a+b)}\) uses the quotient of powers rule to rewrite the expression as \(\frac{p^1}{p^{a+b}}\). This is simply subtracting \((a+b)\) from \(1\). It helps simplify and solve expressions by reducing them to a single power.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (such as additions or multiplications). Understanding how to manipulate these is crucial for solving many mathematical problems. When dealing with exponents in algebraic expressions, particular rules like the product or quotient of powers are used to simplify or rewrite the expression.Here are some key aspects to remember:

• Variables can be raised to powers, which indicates repeated multiplication of the base.
• Always pay attention to the signs and operations within the expression.
• Exponents can sometimes make expressions appear more complex, but applying exponent rules correctly can significantly simplify them.
• Ensure each variable expressed involves positive values, as indicated in the assumptions of many exercises.

In the exercise, \(p^{1-(a+b)}\) is an algebraic expression that can be simplified using exponent rules. Break down and understand each term's role in order to apply the rules effectively. Mastery of these concepts enables the transformation from one form to another easily, providing clarity and simplicity.