Problem 113 - College Algebra

Question

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(\frac{x^{-\frac{5}{4} y^{\frac{1}{3}}}}{x^{-\frac{3}{4}}}\right)^{-6} $$

Step-by-Step Solution

Answer

The simplified form the given expression is \(x^3 y^{-2}\).

Step 1: Apply the power of a quotient rule

The expression is elevated to the power -6. Therefore, this exponent applies to both the numerator and the denominator of the fraction. This is according to the power of a quotient rule that states: \( (\frac{a}{b})^n = \frac{a^n}{b^n} \). So, our expression will be rewritten as: \(\frac{x^{-\frac{5}{4}* -6 } y^{\frac{1}{3} * -6}}{x^{-\frac{3}{4}* -6}} \).

Step 2: Simplify the exponents

Now we simplify each exponent by carrying out the multiplication. Thus we get: \(\frac{x^{\frac{15}{2}} y^{-2}}{x^{\frac{9}{2}}} \).

Step 3: Apply the quotient of powers rule

Now the denominator and the numerator have terms with the same base (x). As per the quotient of powers rule, to divide terms with the same base, subtract the exponents: \( x^{\frac{15}{2}- \frac{9}{2}} = x^{\frac{6}{2}} = x^3 \). So, we have our simplified expression as: \(x^3 y^{-2}\).

Key Concepts

Power of a Quotient Rule

The power of a quotient rule helps simplify expressions where a fraction is raised to an exponent. This rule is crucial because it evenly distributes the power across both the numerator and the denominator.

The rule is expressed as \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). When simplifying, each part of the fraction is raised to the power separately. This step is vital for keeping your calculations organized and ensuring accuracy.

For instance, in our problem, we apply this specific rule by adding the exponent -6 separately to both the numerator and the denominator. Thus, \( \left(\frac{x^{-\frac{5}{4}y^{\frac{1}{3}}}}{x^{-\frac{3}{4}}}\right)^{-6} \) becomes \( \frac{x^{-\frac{5}{4} \cdot -6} y^{\frac{1}{3} \cdot -6}}{x^{-\frac{3}{4} \cdot -6}} \).

Remember that this step sets the stage for easier simplification in subsequent steps of any problem involving fractions and exponents.

Simplifying Expressions

Simplifying expressions involves reducing expressions to their simplest form. This makes complex calculations more manageable and easier to understand.

Once we apply the power of a quotient rule, the next task is to simplify further by actually computing the new exponents. For example, with our expression: \( \frac{x^{\frac{15}{2}} y^{-2}}{x^{\frac{9}{2}}} \),

Here are the steps to simplify it:

• Multiply the exponents as shown, \( -\frac{5}{4} \times -6 = \frac{15}{2} \) and \( \frac{1}{3} \times -6 = -2 \); thus simplifying the powers step by step ensures you keep things systematic.
• This simplification prepares the expression for the next stage, which usually involves applying other exponent rules like the quotient of powers rule.

Following these steps means we're halfway through solving our initial expression.

Quotient of Powers Rule

The quotient of powers rule provides a straightforward method for dividing expressions with the same base. According to this rule, the quotient of two identical bases raised to any exponents is equal to the base raised to the difference of the exponents.

Formally, this is written as: \( \frac{a^m}{a^n} = a^{m-n} \). This rule is essential when simplifying expressions because it helps consolidate terms with the same base.

In our specific example, we have \( \frac{x^{\frac{15}{2}}}{x^{\frac{9}{2}}} \). Using the quotient of powers rule, we subtract the exponent in the denominator from the exponent in the numerator, resulting in \( x^{\frac{15}{2} - \frac{9}{2}} = x^{\frac{6}{2}} = x^3 \).

By effectively applying this rule, we achieve a more simplified version of the expression, leading us to \( x^3 y^{-2} \). This final step completes the process of simplifying the original fraction raised to a negative power.