Problem 2 - Basic Engineering Mathematics

Question

Find the value of: (a) \(\frac{7^{5}}{7^{3}}\) and (b) \(\frac{5^{7}}{5^{4}}\)

Step-by-Step Solution

Answer

(a) 49; (b) 125.

Step 1: Understand the Problem

We need to simplify two expressions: (a) \(\frac{7^{5}}{7^{3}}\) and (b) \(\frac{5^{7}}{5^{4}}\). To solve these expressions, we'll use the properties of exponents.

Step 1: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that when you divide like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Step 2: Simplify Expression (a)

For \(\frac{7^{5}}{7^{3}}\), using the quotient rule, we get \(7^{5-3} = 7^{2}\).

Step 3: Simplify Expression (b)

For \(\frac{5^{7}}{5^{4}}\), similarly, apply the quotient rule to get \(5^{7-4} = 5^{3}\).

Step 4: Compute the Final Values

Calculate the simplified values: \(7^2 = 49\) and \(5^3 = 125\).

Key Concepts

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental principle in mathematics that simplifies division involving powers with the same base. This rule is essential when handling expressions featuring exponents, as it makes otherwise daunting problems much more manageable. According to the quotient rule, when you divide two numbers with the same base, you simply subtract the exponent of the denominator from the exponent of the numerator. This can be expressed in the formula:

• \(\frac{a^m}{a^n} = a^{m-n}\)

The beauty of this rule lies in its straightforwardness and efficiency. If you have \(7^5\) divided by \(7^3\), you apply the rule and simply subtract 3 from 5, resulting in \(7^{2}\). Understanding and applying this rule will definitely save you time and effort when working with exponents.

Properties of Exponents

Exponents have various properties that simplify handling expressions involving them. These properties ensure that calculations remain consistent and logical. In addition to the quotient rule, there are several other properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent Rule: \(a^0 = 1\) for any \(aeq0\).
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\).

Each of these properties helps in simplifying mathematical expressions and solving equations more efficiently. For example, knowing the product of powers property allows for quick multiplication of terms, while the zero exponent rule demystifies what happens when an exponent is zero. These properties build a foundation for further learning in algebra and higher mathematics.

Simplifying Expressions

Simplifying expressions, particularly those involving exponents, is a critical skill in math. This process transforms complex problems into more manageable forms, which are easier to work with and understand. When simplifying, start by identifying applicable rules or properties of exponents. Use the quotient rule when dividing like bases, just like we did to simplify \(\frac{7^{5}}{7^{3}}\) to \(7^2\) and \(\frac{5^{7}}{5^{4}}\) to \(5^3\). Once you've applied the rules, continue simplifying by calculating any remaining terms. In our examples, \(7^2\) becomes 49 and \(5^3\) becomes 125.Breaking down expressions step-by-step not only helps in finding the exact answer but also improves understanding of mathematical concepts. The more you practice this, the swifter and more accurate you will become, paving your way to mastering algebra!