Problem 9 - Basic Engineering Mathematics

Question

Find the value of \(\frac{2^{3} \times 3^{5} \times\left(7^{2}\right)^{2}}{7^{4} \times 2^{4} \times 3^{3}}\)

Step-by-Step Solution

Answer

The value of the expression is 4.5.

Step 1: Simplify the Numerator and Denominator

First, simplify the numerator and the denominator separately. The numerator is given as \(2^{3} \times 3^{5} \times (7^{2})^{2}\). Using the power of a power property, simplify \((7^{2})^{2}\) to \(7^{4}\). Thus, the numerator becomes \(2^{3} \times 3^{5} \times 7^{4}\). The denominator is already given in its simplest form: \(7^{4} \times 2^{4} \times 3^{3}\).

Step 2: Cancel Like Terms

To simplify the fraction, cancel the common terms between the numerator and the denominator. Both numerator and denominator contain \(7^4\), so this term cancels out completely. This leaves the numerator as \(2^3 \times 3^5\) and the denominator as \(2^4 \times 3^3\).

Step 3: Simplify Remaining Terms

Now address the remaining terms: \(\frac{2^3 \times 3^5}{2^4 \times 3^3}\). For the base 2, subtract the exponents: \(2^{3-4} = 2^{-1}\). For the base 3, subtract the exponents: \(3^{5-3} = 3^{2}\). Thus, the expression simplifies to \(\frac{3^2}{2^1} = \frac{9}{2}\).

Step 4: Final Calculation

Perform the final calculation to confirm the simplified result: \(9\div 2 = 4.5\). Therefore, the value of the original expression is \(4.5\).

Key Concepts

Exponent Rules

Understanding exponent rules is essential for working with expressions involving powers. Exponents are a shorthand way to show how many times a number, known as the base, is multiplied by itself. Let's go over some fundamental rules:

• **Product of Powers Rule**: When multiplying two powers with the same base, you add the exponents. For example, for the base 3 in the expression \(3^5 \times 3^3\), you would compute the product as \(3^{5+3} = 3^8\).
• **Power of a Power Rule**: This rule states that when you raise a power to another power, you multiply the exponents. In the expression \((7^2)^2\), you apply the rule to get \(7^{2 \times 2} = 7^4\).
• **Quotient of Powers Rule**: For division, when you divide two powers with the same base, subtract the exponents. So, if you have \(2^3/2^4\), this simplifies to \(2^{3-4} = 2^{-1}\).
• **Negative Exponent Rule**: A negative exponent indicates that the base is on the wrong side of a fraction, and it can be rewritten as a reciprocal. For instance, \(2^{-1}\) becomes \(1/2\).

Knowing these rules allows you to manipulate and simplify expressions efficiently. They were essential in the solution of the given problem, particularly in Step 3, where the exponents of like terms were subtracted.

Simplification Techniques

Simplification of expressions is a critical aspect of making complex problems manageable. The main aim is to reduce expressions to their simplest forms, facilitating elegant and easy calculations. Here are some techniques:

• **Identify Common Factors**: Start by identifying and canceling out common factors in the numerator and denominator of a fraction, such as the \(7^4\) term in both expressions of the original problem. This step simplifies the fraction immediately.
• **Use Exponent Rules**: As depicted in the previous section, employing exponent rules like subtracting exponents when dividing terms helps streamline the problem. For instance, simplifying \(\frac{3^5}{3^3}\) gives \(3^2\).
• **Break Down Complex Expressions**: Use properties such as the power of a power to deal with polynomials or powers within parentheses, thus converting a complex term into a more manageable form. In our example, converting \((7^2)^2\) to \(7^4\) was part of the simplification process.

Through these techniques, making use of mathematical properties allows students to simplify and solve problems with confidence.

Mathematics Education

Mathematics education focuses not just on solving problems but also on developing a deeper understanding of key concepts and techniques. This involves enhancing comprehension through problem-solving exercises like the ones provided in textbook exercises. Here are some tips to ensure effective learning:

• **Conceptual Understanding**: Beyond memorizing rules and steps, achieving a conceptual understanding of why and how these rules apply is crucial. For instance, understanding why exponents are subtracted during division helps in applying the rule accurately.
• **Practice**: Regular practice with a variety of problems helps solidify these concepts and increase speed and accuracy. Problems should vary in complexity to challenge and build skills progressively.
• **Interactive Learning**: Engaging with exercises in group settings or through online platforms encourages discussion, questions, and collaborative problem-solving. This strategy can significantly boost retention and understanding.
• **Feedback and Reflection**: After solving problems, reviewing the steps and reflecting on mistakes or difficult areas is fundamental for growth in mathematics.

Mathematics education, when approached thoughtfully, lays a strong foundation for advanced studies and fosters analytical thinking skills applicable in many other fields.