Problem 13 - Basic Engineering Mathematics

Question

Simplify \(\frac{7^{-3} \times 3^{4}}{3^{-2} \times 7^{5} \times 5^{-2}}\), expressing the answer in index form with positive indices.

Step-by-Step Solution

Answer

\(\frac{3^6 \times 5^2}{7^8}\)

Step 1: Simplify the Numerator

First, let's simplify the numerator of the fraction: \[ 7^{-3} \times 3^4. \] The terms are already in their simplest form, so no action is needed. However, we will simplify the indices later when combining like bases.

Step 2: Simplify the Denominator

Next, simplify the denominator of the fraction:\[ 3^{-2} \times 7^5 \times 5^{-2}. \]Again, these terms are in their simplest form, but remember that we will be re-writing this expression using the laws of indices.

Step 3: Apply Laws of Indices

Now let's combine the terms in the fraction using laws of indices. Simplify using the rule \(a^m \times a^n = a^{m+n}\):\[\frac{7^{-3} \times 3^4}{3^{-2} \times 7^5 \times 5^{-2}} = \frac{7^{-3}}{7^5} \times \frac{3^4}{3^{-2}} \times 5^2.\]Simplifying each fraction by subtracting exponents:\[7^{-3 -5},\quad 3^{4+2},\quad \text{and keep } 5^2.\]

Step 4: Solve for the Indices

Calculate each exponent:- For \(7\), the new exponent is \(-3 - 5 = -8\), giving us \(7^{-8}\).- For \(3\), the new exponent is \(4 + 2 = 6\), giving us \(3^6\).- For \(5\), it remains \(5^2\).

Step 5: Rewrite with Positive Indices

Finally, express all terms with positive indices by using the rule \(a^{-n} = \frac{1}{a^n}\).\[\frac{3^6 \times 5^2}{7^8}\]

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics, especially when dealing with complex fractions containing exponents. When you encounter an expression like \(\frac{7^{-3} \times 3^{4}}{3^{-2} \times 7^{5} \times 5^{-2}}\), your goal is to rewrite it in a simpler form, which often means combining similar terms and reducing the overall complexity.
To simplify such expressions:

• Identify terms that share the same base.
• Utilize the laws of indices to combine these terms.

For example, in the expression provided, the numerator and denominator each have terms with bases of 7, 3, and 5. By applying the laws of indices, you combine these terms and simplify the expression overall. Resulting in fewer terms and lower exponents, making the expression easier to work with or interpret. Breaking the problem into smaller parts facilitates simpler arithmetic handling and reduces human error.

Exponents Management

Exponents management involves handling and simplifying different powers or indices within a mathematical expression or equation. They determine how many times a base number is multiplied by itself.
Here are some fundamental principles for managing exponents:

• Multiplying like bases: When multiplying terms with the same base, add the exponents, as in the rule \(a^m \times a^n = a^{m+n}\).
• Dividing like bases: When dividing terms with the same base, subtract the exponents, following \(\frac{a^m}{a^n} = a^{m-n}\).
• Negative exponents: A negative exponent \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\).

In our example, the legal manipulation of exponents is the application of these principles: 7 with exponents \(-3 - 5 = -8\), 3 with exponents \(4 + 2 = 6\), and leaving 5's exponent as \(5^2\). Each manipulation relies on the consistent use of these rules to simplify and solve the equation effectively.

Index Notation

Index notation, or exponential notation, is a mathematical shorthand used to express repeated multiplication of the same number. This form is particularly useful for simplifying and interpreting large numbers and complex expressions. It is crucial for students to understand this notation for ease in understanding further mathematical concepts.
Some pointers to grasp index notation:

• An expression like \(a^n\) signifies "a" raised to the power of "n," indicating that "a" is multiplied by itself \(n\) times.
• Positive indices, like \(a^2\) or \(a^3\), simplify expressing large repeated multiplications.
• Negative indices indicate the reciprocal of the base raised to the corresponding positive index, such as \(a^{-n} = \frac{1}{a^n}\).

The final expression \(\frac{3^6 \times 5^2}{7^8}\) is an example of index notation used to simplify and understand a problem more fundamentally. Here, the indices are positive, achieving a goal of clarity and simplification in mathematical communication.