Problem 589 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) (b) \(\frac{n^{\frac{3}{7}} \cdot n}{n^{-\frac{4}{7}}}\)

Step-by-Step Solution

Answer

(a) m, (b) n^2

Step 1: Simplify the exponents in the numerator (Part a)

Combine the exponents in the numerator using the property: \(a^m \times a^n = a^{m+n}\). \(m^{\frac{7}{4}} \times m^{-\frac{5}{4}} = m^{\frac{7}{4} - \frac{5}{4}} = m^{\frac{2}{4}}\).

Step 2: Simplify the fraction (Part a)

Use the same base exponent property to simplify the fraction: \( \frac{m^{\frac{2}{4}}}{m^{-\frac{2}{4}}} = m^{\frac{2}{4} - (-\frac{2}{4})} = m^{\frac{2}{4} + \frac{2}{4}} = m^{\frac{4}{4}} = m^1 = m\).

Step 3: Simplify the exponents in the numerator (Part b)

Combine the exponents in the numerator: \(n^{\frac{3}{7}} \times n = n^{\frac{3}{7}} \times n^{1} = n^{\frac{3}{7} + 1} = n^{\frac{3}{7} + \frac{7}{7}} = n^{\frac{10}{7}}\).

Step 4: Simplify the fraction (Part b)

Use the same base exponent property to simplify the fraction: \( \frac{n^{\frac{10}{7}}}{n^{-\frac{4}{7}}} = n^{\frac{10}{7} - (-\frac{4}{7})} = n^{\frac{10}{7} + \frac{4}{7}} = n^{\frac{14}{7}} = n^2\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions correctly. Exponential expressions follow certain rules that make the simplification process systematic and straightforward. The properties you will encounter often include:

• Product of Powers: For any base 'a', when multiplying powers of the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).


• Quotient of Powers: For the same base 'a', when dividing powers, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).


• Power of a Power: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

These properties help simplify complex expressions and resolve fractional, negative, or even zero exponents. Always remember to handle the exponents carefully, applying the relevant rule based on whether you are multiplying, dividing, or raising powers.

Fractional Exponents

Fractional exponents are another way to express roots. Instead of writing \(\sqrt[n]{a}\), you can write it using fractional exponents as \(a^{\frac{1}{n}}\). Here are some key pointers for working with fractional exponents:

• A fractional exponent \(a^{\frac{m}{n}}\) means you take the nth root of 'a' and then raise it to the mth power. This can be written as \(\sqrt[n]{a^m}\).


• Combining fractional exponents uses the same properties of exponents as integer exponents. For example, \(a^{\frac{1}{2}} \times a^{\frac{1}{3}} = a^{\frac{1}{2} + \frac{1}{3}}\).
• To simplify expressions with fractional exponents, apply the properties of exponents methodically and be careful with the arithmetic of the fractions.

Consider the exercise given: \( \frac{m^{\frac{7}{4}} \times m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}} \). By applying the product and quotient rules, you combine and reduce the exponents step by step, leading to a simpler expression.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression as simple as possible by combining like terms, removing parentheses, and reducing fractions. Let's break this down:

• Combine Like Terms: Only terms with the same base and exponent can be combined. For example, \(m^{\frac{7}{4}} \times m^{-\frac{5}{4}}\) results in \( m^{\frac{2}{4}} \). Adding or subtracting different exponents follows the same arithmetic rules: \( \frac{2}{4} - ( - \frac{2}{4} ) = \frac{4}{4}\).
• Removing Parentheses: Use the distribution property and properties of exponents to simplify expressions inside the parentheses first and then eliminate them when possible.
• Reducing Fractions: Always seek to reduce fractions to their simplest forms. In the given exercise, the final steps involve simplifying \( \frac{m^{2/4}}{m^{-2/4}} = m^{1} \) and \( \frac{n^{10/7}}{n^{-4/7}} = n^{2} \).

Remember, algebraic simplification often requires patience and practice. The more exercises you do, the more familiar you will become with the techniques and properties you need to apply.