Problem 47 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. $$ \left(\frac{m^{10}}{n}\right)^{8} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{m^{80}}{n^8}\).

Step 1: Apply the Power of a Quotient Rule

When raising a fraction to a power, apply the power to both the numerator and the denominator separately. Thus, the expression \(\left(\frac{m^{10}}{n}\right)^{8}\) becomes \(\frac{(m^{10})^8}{n^8}\).

Step 2: Apply the Power Rule for Exponents

Use the power rule \((a^m)^n = a^{m \cdot n}\) to simplify the numerator. Here, \((m^{10})^8 = m^{10 \cdot 8} = m^{80}\). Therefore, the expression simplifies to \(\frac{m^{80}}{n^8}\).

Step 3: Final Simplified Expression

The expression \(\left(\frac{m^{10}}{n}\right)^{8}\) simplifies to \(\frac{m^{80}}{n^8}\) with positive exponents in both the numerator and the denominator.

Key Concepts

Power of a Quotient Rule

The Power of a Quotient Rule is a fundamental rule in exponentiation that allows you to simplify expressions where a fraction is raised to a power. This rule helps break down more complex expressions into simpler forms that are easier to work with.

When you have a fraction like \( \left( \frac{a}{b} \right)^n \), you apply the power to both the numerator and the denominator separately. That means \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).

• Each part of the fraction gets raised individually to the power given.
• This rule helps maintain the balance of the expression.

Applying this rule to our original expression \( \left( \frac{m^{10}}{n} \right)^8 \), we get \( \frac{(m^{10})^8}{n^8} \). Here, both \( m^{10} \) and \( n \) are raised to the 8th power, showing how the entire fraction is affected by the power.

Power Rule for Exponents

The Power Rule for Exponents is an essential exponentiation law used to simplify expressions where an exponent is raised to another exponent. This rule states that when you have an expression such as \((a^m)^n\), you can simplify it by multiplying the exponents, making it \(a^{m \cdot n}\).

This rule allows for straightforward simplification:

• It reduces complex exponentiation into basic multiplication.
• By multiplying the exponents, we simplify the expression significantly.

For our expression \((m^{10})^8\), we apply this rule. Multiply 10 by 8 to obtain the expression \(m^{80}\). This simplification shows how powerful this rule is in reducing complicated expressions into manageable parts.

Simplifying Expressions

Simplifying expressions is the core objective when dealing with mathematical problems involving exponents. The aim is to transform the expression into its simplest form while ensuring it remains equal in value.

Here are some key points about simplifying:

• Use rules of exponents to reduce the complexity.
• Ensure that the final form has positive exponents where possible.
• Combine like terms when applicable.

In our example, we start with the expression \(\left(\frac{m^{10}}{n}\right)^8\). By applying both the Power of a Quotient and Power Rule for Exponents, we simplify it to \(\frac{m^{80}}{n^8}\). This final expression has positive exponents and is expressed in the simplest form, making it easier to understand and work with.