Problem 21 - Basic Engineering Mathematics

Question

Simplify \(\frac{\left(m n^{2}\right)^{3}}{\left(m^{1 / 2} n^{1 / 4}\right)^{4}}\)

Step-by-Step Solution

Answer

The simplified expression is \(mn^5\).

Step 1: Simplify the Numerator

The numerator is \((mn^2)^3\). To simplify this, we apply the power rule for exponents, which states that \((a^b)^c = a^{b\times c}\). Thus, \((mn^2)^3 = m^3(n^2)^3 = m^3n^{6}\).

Step 2: Simplify the Denominator

The denominator is \((m^{1/2} n^{1/4})^4\). Using the same power rule for exponents, we have \(m^{1/2}\times4 = m^2\) and \(n^{1/4}\times4 = n\). Therefore, the denominator simplifies to \(m^2n\).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

We have \(\frac{m^3n^6}{m^2n}\). To simplify this division, use the quotient rule for exponents \(\frac{a^b}{a^c} = a^{b-c}\). For \(m\), we have \(m^{3-2} = m\), and for \(n\), we have \(n^{6-1} = n^5\).

Step 4: Conclusion

Thus, the simplified form of the expression is \(mn^5\).

Key Concepts

Power Rule for Exponents

When it comes to simplifying expressions involving exponents, the Power Rule for Exponents is one of the essential tools in our math toolbox. This rule is particularly helpful when you encounter a power raised to another power. The power rule states that \( (a^b)^c = a^{b \times c} \). This means you multiply the exponents together.

In the exercise above, we applied the power rule to both the numerator and the denominator. For the numerator \( (mn^2)^3 \), you apply the power rule to each component: \( m^3 \) and \( (n^2)^3 \), yielding \( m^3n^6 \). In like manner, the denominator \( (m^{1/2} n^{1/4})^4 \) follows the same principle: \( m^2n\).

Understanding the power rule is crucial because it simplifies complex expressions, making it easier to deal with each part. Once you've mastered it, simplifying even the most intimidating expressions becomes straightforward.

Quotient Rule for Exponents

The Quotient Rule for Exponents is another essential rule that aids in simplifying expressions, especially those involving division. The rule states that if you divide powers with the same base, you can subtract the exponents: \( \frac{a^b}{a^c} = a^{b-c} \).

In our exercise solution, after simplifying the numerator and the denominator separately, we applied the quotient rule to the expression \( \frac{m^3n^6}{m^2n} \). By subtracting the exponents of the common bases, we obtain \( m^{3-2} \) and \( n^{6-1} \). This simplifies to \( mn^5 \).

Using the quotient rule correctly can drastically simplify expressions, as it allows you to work through complicated fractions in a streamlined manner. Remember, this rule only applies when the bases are identical, so always watch for this to avoid mistakes.

Numerator and Denominator Simplification

When dealing with expressions with both a numerator and a denominator, it is important to simplify each part before proceeding to other steps. Simplifying these components separately makes it easier to identify possible further simplifications.

For the given exercise, the numerator \( (mn^2)^3 \) was simplified to \( m^3n^6 \) using the power rule. Simultaneously, the denominator \( (m^{1/2} n^{1/4})^4 \) was also reduced to \( m^2n \).

By breaking down each part, you reduce the complexity of the expression. This systematic simplification prepares the expression for final adjustments using additional rules like the quotient rule, bringing you step-by-step to the final answer. This approach is not only efficient but also reduces the risk of errors in your calculations.