Problem 8 - Basic Engineering Mathematics

Question

Evaluate \(\frac{3^{3} \times 5^{7}}{5^{3} \times 3^{4}}\)

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{625}{3} \).

Step 1: Write Down the Expression

The expression we need to evaluate is \( \frac{3^3 \times 5^7}{5^3 \times 3^4} \). This is a fraction with bases raised to powers in both the numerator and the denominator.

Step 2: Simplify Powers of the Same Base

We separate and simplify the powers of the same base: - For base 3: \( \frac{3^3}{3^4} = 3^{3-4} = 3^{-1} \). - For base 5: \( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \). This uses the property \( \frac{a^m}{a^n} = a^{m-n} \).

Step 3: Combine Simplified Terms

Combine the simplified results from the previous step: \( \frac{3^3 \times 5^7}{5^3 \times 3^4} = 3^{-1} \times 5^4 \).

Step 4: Evaluate Numerical Expression

Now compute the numerical values for the simplified expression: - \(3^{-1} = \frac{1}{3}\),- \(5^4 = 625\).Thus, the expression becomes \( \frac{1}{3} \times 625 = \frac{625}{3} \).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves making them easier to read and solve without changing their value. When you encounter expressions with multiple terms or exponents, the goal is to use algebraic rules to reduce these expressions to their simplest form.

Here are some key points to remember when simplifying expressions:

• Identify like terms, which are terms with the same variable raised to the same power, and combine them.
• Apply the rules of exponents to simplify terms that have the same base, using the properties such as \(a^m \times a^n = a^{m+n}\) or \(\frac{a^m}{a^n} = a^{m-n}\).
• Factorization might also help in simplifying by rewriting complex expressions in their factor forms.

These strategies make it easier to work through complex algebra problems and lead to quicker solutions.

Fractional Powers

Fractional powers, also known as rational exponents, represent both an exponent and a root. The term \(a^{m/n}\) signifies that we take the \(n\)-th root of \(a\) raised to the \(m\)-th power. This can sometimes be confusing, but breaking it into steps helps.

Consider the expression \(a^{3/2}\):

• First, take the square root of \(a\), which is \(a^{1/2}\).
• Then, raise this result to the power of 3.

Here’s how to think of it in terms of the operation:

• The denominator of the fraction (2 in this case) indicates the root you should take.
• The numerator (3 here) shows the power to raise that root to afterward.

This concept is powerful and extends naturally from integers to rational numbers, offering a deeper way to understand exponential calculations.

Basic Algebraic Manipulation

Basic algebraic manipulation involves using algebraic properties to rearrange or simplify expressions and solve equations. This is a core skill in algebra for transforming expressions into a form where the desired solution or simplification becomes apparent.

Some of the fundamental rules for manipulation include:

• Distributive Property: When you have to multiply a term by a sum or difference, distribute the term to each part within the parentheses, like \(a(b + c) = ab + ac\).
• Associative Property: You can regroup terms without changing the result, as in \((a + b) + c = a + (b + c)\).
• Commutative Property: You can change the order of terms being added or multiplied, i.e., \(a + b = b + a\) or \(ab = ba\).

Applying these rules helps simplify expressions and makes it easier to solve or evaluate them. By keeping these properties in mind, you can tackle a wide range of algebraic problems efficiently.