Problem 575 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}}\) (b) \(\frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}}\) c \(\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)

Step-by-Step Solution

Answer

a) s, b) z^2, c) \(\frac{1}{w}\)

Step 1: Identify the given fractions

We have three fractions with exponents to simplify: (a) \( \frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}} \) (b) \( \frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}} \) (c) \( \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} \)

Step 1: Apply the quotient rule of exponents for (a)

For (a), use the quotient rule of exponents \( \frac{a^m}{a^n} = a^{m-n} \). Therefore, \[ \frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}} = s^{\frac{11}{5} - \frac{6}{5}} = s^{\frac{5}{5}} = s^1 = s \]

Step 2: Apply the quotient rule of exponents for (b)

For (b), similarly, use the quotient rule of exponents. \[ \frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}} = z^{\frac{7}{3} - \frac{1}{3}} = z^{\frac{6}{3}} = z^2 \]

Step 3: Apply the quotient rule of exponents for (c)

For (c), use the quotient rule of exponents. \[ \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = w^{\frac{2}{7} - \frac{9}{7}} = w^{-\frac{7}{7}} = w^{-1} = \frac{1}{w} \]

Step 5: Simplified results

After simplifying each fraction, we get: (a) \(s\) (b) \(z^2\) (c) \(\frac{1}{w}\)

Key Concepts

Quotient Rule of Exponents

To simplify expressions with exponents, we can use the quotient rule. The rule states: \[ \frac{a^m}{a^n} = a^{m-n} \] This means that when you divide like bases, you subtract the exponents.

For example, if you have \( \frac{x^5}{x^3} \), you subtract 3 from 5 to get \( x^{5-3} = x^2 \). The key is to subtract the exponent of the denominator from the exponent of the numerator.

Using this rule helps simplify complicated algebraic expressions, making them easier to manage.

Simplifying Expressions

Simplifying expressions means making them as simple as possible. It often involves multiple steps: applying rules like the quotient rule of exponents, combining like terms, and canceling common factors.

Let's take the expression \( \frac{a^7}{a^2} \). Using the quotient rule of exponents, we get \( a^{7-2} = a^5 \). Another example: \( \frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}} \). \[ \frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}} = s^{\frac{11}{5} - \frac{6}{5}} = s^{\frac{5}{5}} = s^1 = s \]

Simplified forms are easier to understand and work with in further calculations.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. Simplifying these fractions helps in solving algebra problems.

Consider the fraction \( \frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}} \). By applying the quotient rule: \[ \frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}} = z^{\frac{7}{3} - \frac{1}{3}} = z^{\frac{6}{3}} = z^2 \] Another example is \( \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} \). Applying the quotient rule, we get \[ \frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}} = w^{\frac{2}{7} - \frac{9}{7}} = w^{-\frac{7}{7}} = w^{-1} = \frac{1}{w} \]

Simplifying algebraic fractions like this makes solving equations faster and reduces errors.