Problem 2 - Basic Engineering Mathematics

Question

Find the value of \(4 p^{2} q r^{3}\), given that \(p=2\), \(q=\frac{1}{2}\) and \(r=1 \frac{1}{2}\)

Step-by-Step Solution

Answer

The value is 27.

Step 1: Substitute Given Values into the Expression

The given expression is \( 4p^{2}qr^{3} \). Start by substituting the given values: \( p = 2 \), \( q = \frac{1}{2} \), and \( r = 1 \frac{1}{2} \) into the expression.

Step 2: Convert Mixed Number to an Improper Fraction

Convert the mixed number \( r = 1 \frac{1}{2} \) into an improper fraction. We do this by multiplying 1 by 2 and adding 1: \( r = \frac{3}{2} \).

Step 3: Simplify the Expression \, \( 4 (2)^2 \frac{1}{2} \left( \frac{3}{2} \right)^3 \)

Plugging all substitutions into the expression gives us \( 4 \times (2)^2 \times \frac{1}{2} \times \left( \frac{3}{2} \right)^3 \). Calculate each part to simplify.

Step 4: Calculate \( (2)^2 \)

Calculate \( (2)^2 = 4 \).

Step 5: Calculate \( \left( \frac{3}{2} \right)^3 \)

Calculate the cube of \( \frac{3}{2} \). Raise both numerator and denominator separately: \( \left( \frac{3}{2} \right)^3 = \frac{27}{8} \).

Step 6: Insert and Calculate Full Expression

Place all calculated terms back into the expression: \( 4 \times 4 \times \frac{1}{2} \times \frac{27}{8} \).

Step 7: Calculate \( 4 \times 4 \)

Perform multiplication: \( 4 \times 4 = 16 \).

Step 8: Simplify \( 16 \times \frac{1}{2} \)

Calculate: \( 16 \times \frac{1}{2} = 8 \).

Step 9: Final Multiplication with Fraction

Finally, multiply the remaining numbers: \( 8 \times \frac{27}{8} \). The \( 8 \) in the numerator and denominator cancel out, resulting in \( 27 \).

Key Concepts

Substitution

Substitution is a fundamental algebraic technique where given values are placed into an expression or equation in place of their respective variables. This helps in evaluating or simplifying expressions. Here's how it works:

• Take the expression you need to solve. In our case, it is the expression for multiplying and raising variables: \(4p^{2}qr^{3}\).
• Identify the values for each variable. We have \(p = 2\), \(q = \frac{1}{2}\), and \(r = 1\frac{1}{2}\).
• Replace each variable in the expression with these known values. This gives you a new expression composed entirely of numbers instead of letters.

Using substitution transforms an algebra problem into a numerical problem. It eliminates variables and allows you to directly work with numbers.
Keep in mind that the order of operations must still apply during substitution. First, substitute the values and then proceed with any exponents, multiplication, and division as you normally would.

Simplification of Expressions

Simplification of expressions involves reducing mathematical expressions down to the simplest form possible. This process is all about making the expression easier to grasp without changing its value.

• Begin with substitution as described above, resulting in numerical expressions.
• Apply the proper order of operations. First calculate any exponents (like \((2)^2 = 4\) or \(\left( \frac{3}{2} \right)^3 = \frac{27}{8}\)).
• Next, carry out multiplications (such as \(4 \times 4 = 16\)) and any divisions.
• Combine all results to simplify the expression further. Continuously reduce until no further simplification is possible.

By systematically simplifying, you make complex expressions more manageable, ultimately revealing a neatly solved form to understand and use in further calculations.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator. They can arise from mixed numbers by conversion. It’s sometimes easier to work with improper fractions.To convert a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator (e.g., \(1 \times 2 = 2\)).
• Add this result to the numerator of the fraction (e.g., \(2 + 1 = 3\)).
• The result becomes the new numerator over the original denominator, forming an improper fraction (\(\frac{3}{2}\) from \(1 \frac{1}{2}\)).

Improper fractions simplify multiplication and division and help maintain consistency in numeric expressions. They are particularly useful in simplifying expressions that involve powers.