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Chapter 6: Problem 390

Simplify. $$\left(\frac{p}{r^{11}}\right)^{2}$$

Short Answer

Expert verified
\frac{p^{2}}{r^{22}}

Step by step solution

01

Understand the Problem

The problem requires simplifying the expression \(\frac{p}{r^{11}}\)^{2}.
02

Apply the Power of a Quotient Rule

The rule states that \(\frac{a}{b}\)^{n} = \(\frac{a^{n}}{b^{n}}\). So, apply this rule to the given expression: \(\frac{p}{r^{11}}\)^{2} = \(\frac{p^{2}}{(r^{11})^{2}}\).
03

Simplify the Denominator

Simplify \(r^{11}\)^{2} by multiplying the exponents: \(r^{11 \times 2} = r^{22}\).
04

Write the Final Simplified Expression

Combine the results from the previous steps: \(\frac{p^{2}}{r^{22}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Quotient Rule
When you encounter an expression of the form \(\frac{a}{b}\right)^{n}\), you can apply the Power of a Quotient rule. This rule tells us that raising a quotient to a power is the same as raising both the numerator and the denominator to that power separately. So, \(\frac{a}{b}\right)^{n}\) equals \(\frac{a^{n}}{b^{n}}\). This simplifies your computation by allowing you to handle each part of the fraction independently.
In the given exercise, \(\frac{p}{r^{11}}\right)^{2}\), we use this rule to separate the powers: \[ \frac{p}{r^{11}} \right)^{2} = \frac{p^{2}}{(r^{11})^{2}} \right.\right) \]
This makes the expression easier to manage.
Exponent Multiplication
Exponent multiplication rules are crucial for simplifying expressions. When you raise a power to another power, you multiply the exponents. Mathematically, \( (a^{m})^{n} = a^{m \times n} \). This step helps simplify expressions that have variables raised to an exponent which is then raised to another exponent.
For instance, in the step \( (r^{11})^{2} = r^{11 \times 2} = r^{22} \), we are applying the exponent multiplication rule. Each exponentiation amplifies the effect of the original exponent.
This rule streamlines the expression and makes it look less complicated.
Algebraic Simplification
Algebraic simplification involves performing operations that make an expression easier to understand. This can include steps like combining like terms, reducing fractions, or applying exponent rules.
In our example, after you've applied the Power of a Quotient rule and simplified with exponent multiplication, you're left with a cleaner expression: \( \frac{p^{2}}{r^{22}} \). This final expression is more straightforward and easier to interpret.
Simplifying expressions is about making them as simple and clear as possible, without changing their value. This process helps you solve algebraic problems more efficiently and accurately.

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