Problem 38 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \left(\frac{m^{3}}{n^{?}}\right)^{4}=\frac{m^{12}}{n^{8}} $$

Step-by-Step Solution

Answer

The missing exponent in the denominator is 2.

Step 1: Identify the given equation

The given equation is \( \left( \frac{m^{3}}{n^{?}} \right)^{4} = \frac{m^{12}}{n^{8}} \)

Step 2: Apply the power of the power rule

The power of a power rule states that \((a^m)^n = a^{mn}\). When we apply the power of a power rule to the left side of the equation, we have \( m^{3*4} = m^{12} \) and \( n^{?*4} = n^{8} \). The first equation \( m^{3*4} = m^{12} \) is valid, hence our exponent for m is correct. Now we must solve \( ?*4 = 8 \) by dividing both sides by 4.

Step 3: Solve for the unknown exponent

We want to solve \( ?*4 = 8 \). To isolate ?, we must divide both sides of the equation by 4. Therefore, ? = 8/4 = 2.

Key Concepts

Power of a Power Rule

Understanding the power of a power rule is a key concept in working with exponents. When we deal with an expression like \((a^m)^n\), we apply this rule to simplify it.
Instead of dealing with powers over powers, the rule tells us that we just need to multiply the exponents: \((a^m)^n = a^{mn}\).
This rule makes it easy to handle nested exponents.

• If you have \(x^2\) raised to the 3rd power, it becomes \((x^2)^3 = x^{2 \times 3} = x^6\).
• It simplifies expressions and helps in problem-solving scenarios.

In the context of the given exercise, applying this rule helps us determine how the exponents on both the numerator and the denominator should be manipulated to match the desired outcome.

Solving Equations

Solving equations with exponents involves finding the unknowns that make the equation true. First, identify all parts of the equation and apply any relevant rules to simplify terms.
In the exercise, the task is to find the missing exponent represented by the question mark.

• We focus on the relationship \(n^{?*4} = n^8\).
• By dividing both sides by 4, we solve for the unknown: \(? \times 4 = 8\), leading to \(? = \frac{8}{4} = 2\).

The key to solving equations with exponents is consistency and ensuring operations on both sides are balanced. Work through the equation step-by-step to isolate the unknown and find a solution.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. They are a fundamental part of algebra and appear in various mathematical contexts.
For instance, \(m^3\) means \(m\) multiplied by itself twice (\(m \times m \times m\)).

• Exponents can alter expressions significantly, so it’s crucial to understand how they work.
• Negative exponents, like \(m^{-3}\), represent the reciprocal: \(\frac{1}{m^3}\).
• Zero as an exponent, \(m^0\), equals 1, no matter the base, given \(m eq 0\).

Understanding how to manipulate and work with exponents is essential, as they are foundational for solving more complex mathematical problems. They help simplify equations and bring efficiency to calculations.