Problem 10 - Basic Engineering Mathematics

Question

Evaluate: \(\frac{4^{1.5} \times 8^{1 / 3}}{2^{2} \times 32^{-2 / 5}}\)

Step-by-Step Solution

Answer

The value of the expression is 16.

Step 1: Simplify Individual Terms

First, let's simplify each term separately. For \(4^{1.5}\), this can be written as \(4^{3/2} = (2^2)^{3/2} = 2^{3}\). Hence, \(4^{1.5} = 8\).For \(8^{1/3}\), this can be written as \((2^3)^{1/3} = 2^{3/3} = 2^{1}\). Hence, \(8^{1/3} = 2\).For \(2^2\), it remains as \(4\).For \(32^{-2/5}\), this can be written as \((2^5)^{-2/5} = 2^{-2}\). Hence, \(32^{-2/5} = \frac{1}{4}\).

Step 2: Substitute Simplified Terms into the Expression

Substituting the simplified terms back into the original expression, we now have:\[\frac{8 \times 2}{4 \times \frac{1}{4}}\]

Step 3: Simplify the Numerator and Denominator

Simplify the numerator:\(8 \times 2 = 16\).Simplify the denominator:\(4 \times \frac{1}{4} = 1\).Now, the expression becomes:\(\frac{16}{1} = 16\).

Key Concepts

Simplifying Expressions

Working with expressions often requires simplifying them to make calculations easier. Simplifying can involve breaking down components, especially when dealing with exponents. For example, in the exercise expression \(\frac{4^{1.5} \times 8^{1 / 3}}{2^{2} \times 32^{-2 / 5}}\), each term can be simplified individually before solving the whole expression.

• For \(4^{1.5}\), recognizing that \(4 = 2^2\) allows us to apply the power rule, converting it to \(2^{3}\).
• Similarly, \(8^{1/3}\) simplifies from \(8 = 2^3\) to \(2^{1}\) using the same power rules.
• The term \(32^{-2/5}\) becomes \((2^5)^{-2/5} = 2^{-2}\), translating it to a simpler form, \(\frac{1}{4}\).

Simplifying expressions in this way makes further calculations much more straightforward.

Fraction Evaluation

Evaluating expressions with fractions is all about simplifying the numerator and the denominator separately to break down the complexity. In our exercise, we've simplified each component:

• Numerator: \(8 \times 2 = 16\).
• Denominator: \(4 \times \frac{1}{4} = 1\).

Once both the numerator and the denominator are in their simplest forms, the fraction \(\frac{16}{1}\) becomes much easier to understand and solve, arriving at a result of 16.

Power Rules

Power rules in mathematics help us manage expressions involving exponents. These rules simplify expressions greatly and are essential when working with roots and powers. Let's break down the key power rules used in this exercise:

• Product of Powers: When you multiply terms with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Fractional Exponents: A fractional exponent like \(a^{m/n}\) means the n-th root of the base raised to the m power, \(\sqrt[n]{a^m}\).
• Negative Exponents: Inverse the base and convert the negative exponent to a positive exponent, \(a^{-m} = \frac{1}{a^m}\).

Understanding and applying these power rules not only aids in simplifying expressions but also makes evaluating them more intuitive and systematic.