Problem 47 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{m p}{n}\right)^{9} $$

Step-by-Step Solution

Answer

\( \frac{m^9 p^9}{n^9} \)

Step 1: Identify the Power Rule

The power rule states that when you raise a quotient to a power, you can apply the power to both the numerator and the denominator separately. Thus, for the expression \( \left(\frac{mp}{n}\right)^9 \), the power of 9 can be distributed to both \( m \), \( p \), and \( n \).

Step 2: Apply the Power Rule to Each Component

Raise the numerator \( mp \) and denominator \( n \) to the power of 9 separately. This gives us: \[ \frac{(mp)^9}{n^9}. \]

Step 3: Apply the Power of a Product Rule

The power of a product rule states that \((ab)^n = a^n b^n\). Apply this rule to the numerator \((mp)^9\):\[ (mp)^9 = m^9 p^9. \]

Step 4: Rewrite the Simplified Expression

Substitute back into the fractional expression to get: \[ \frac{m^9 p^9}{n^9}. \]

Key Concepts

Power of a Product

When you encounter an expression where two or more variables are multiplied together inside a bracket and then raised to a power, you use the power of a product rule. This rule is helpful in simplifying such expressions by breaking them down. Imagine you have \[(a \, b)^n\]This expression represents both variables a and b being multiplied and then elevated to the power n. According to the power of a product rule, you can simply raise each factor inside the brackets to the power n individually. So, \[(a \, b)^n\]becomes \[a^n \, b^n\].
This rule is incredibly useful because it allows you to spread the power over each part of the product separately, which makes it easier to work through complex expressions. It is crucial to keep in mind that the power must apply entirely across to each factor to maintain the expression's original value. When applied to our original exercise, \[(mp)^9\], this becomes \[m^9 \, p^9\].

Quotient Rule

The quotient rule in exponents is often used when you deal with division inside an expression that is raised to a power. Think of it as an extension of the rule you just learned for products. When you have two quantities in a fraction, both inside a power:\[ \left( \frac{a}{b} \right)^n \] you can apply the power to both the numerator and the denominator separately. This means \[(a^n) / (b^n)\].
This rule is significant because it reassures us that raising a quotient to a power won’t change the essence of the operation as long as each part receives the same power. When you apply this rule, it becomes simpler to handle divisions in algebraic expressions raised to a high power without altering their inherent relations. This rule was used to distribute the 9th power to each part in the expression, \[\left(\frac{mp}{n}\right)^9\], resulting in \[\frac{(mp)^9}{n^9}\].

Exponentiation

Understanding exponentiation is key to mastering algebraic manipulations. Exponentiation, in simple terms, means raising a number or a variable to a specific power. It’s a form of repeated multiplication.
When we say\[a^n\],we mean that we multiply a by itself n times. For instance, raising 3 to the power of 2, written as \[3^2\], means \[3 \times 3 = 9\].
In our exercise, when the expression \[\left(\frac{mp}{n}\right)^9\] is simplified, each element \[m\], \[p\], and \[n\] gets raised to the 9th power. Exponentiation helps take complex expressions and make them simpler by dealing with exponential terms individually. It’s important to never assume that multiplication of bases will simplify powers like sums; exponentiation strictly follows its rules.This strong and powerful mathematical tool allows us to express and manage large numbers and complicated algebraic expressions effectively. It ensures that each term is given its due multiplication power.