Problem 111 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ -\left(\frac{a^{4}}{81}\right)^{3 / 4} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{a^3}{27}\).

Step 1: Apply the Power of a Quotient Rule

The expression is \(-\left(\frac{a^{4}}{81}\right)^{3/4}\). Start by applying the power of a quotient rule, which states \(\left(\frac{u}{v}\right)^n = \frac{u^n}{v^n}\). Here, \(u = a^4\), \(v = 81\), and \(n = \frac{3}{4}\). So we have: \(-\frac{(a^4)^{3/4}}{81^{3/4}}\).

Step 2: Simplify the Numerator

Simplify the numerator \((a^4)^{3/4}\) using the power of a power rule, \((x^m)^n = x^{m \times n}\). Thus, \((a^4)^{3/4} = a^{4 \times 3/4} = a^{3}\).

Step 3: Simplify the Denominator

Simplify the denominator \(81^{3/4}\). We know \(81 = 3^4\). So the expression becomes \((3^4)^{3/4}\), which simplifies as \((3^4)^{3/4} = 3^{4\times 3/4} = 3^{3}\). Calculate \(3^3 = 27\).

Step 4: Compose the Simplified Expression

Substitute the simplified numerator \(a^3\) and the simplified denominator \(27\) back into the expression to get \(-\frac{a^3}{27}\). This is the simplified form of the original expression.

Key Concepts

Power of a Quotient Rule

When you encounter an expression in the form of \( \left(\frac{u}{v}\right)^n\), the power of a quotient rule can be really handy. This rule tells us that we can distribute the exponent \(n\) to both the numerator \(u\) and the denominator \(v\).
This means \( \left(\frac{u}{v}\right)^n = \frac{u^n}{v^n} \). The logic here is quite straightforward: any power that affects a fraction applies equally to its numerator and denominator.

• Divide the base of the fraction into numerator \(u\) and denominator \(v\).
• Apply the power \(n\) to both \(u\) and \(v\) individually.
• Write the new fraction \(\frac{u^n}{v^n}\).

In our example, \(u = a^4\), \(v = 81\), and \(n = \frac{3}{4}\), so we have \(-\frac{(a^4)^{3/4}}{81^{3/4}}\). It simplifies the process and separates the expression into two easier to manage pieces.

Power of a Power Rule

Now let's understand what happens when you see an expression like \((x^m)^n\). The power of a power rule shows us a convenient way to handle that. It tells us that when you have a power raised to another power, you can multiply the exponents.
So, \((x^m)^n = x^{m \times n}\). This simplification trick is not only helpful but also essential in making even complex expressions easier to manage.

• Identify the base \(x\) and the powers \(m\) and \(n\).
• Multiply the two powers \(m \times n\).
• Rewrite the base with the new exponent \(x^{m \times n}\).

In our case, using this rule simplifies \((a^4)^{3/4}\) into \(a^{3}\). This step helps us in simplifying the overall expression and getting closer to our simpler form.

Simplifying Expressions

The final step in many algebra problems is simplifying the expression to its shortest form. Simplifying is all about reducing the complexity while maintaining the original value of the expression. This gives us expressions that are not only more practical to use but make patterns and solutions clearer.

• Look for ways to group like terms.
• Apply rules like the power of a quotient or power of a power.
• Remember substitution with known values or simplifying constants.

In our specific example, after applying the rules, the expression \(-\left(\frac{a^{4}}{81}\right)^{3 / 4}\) simplifies down to \(-\frac{a^3}{27}\). This step-by-step procedure not only clarifies the expression but ensures it's as neat and manageable as possible. Simplifying helps present solutions in their clearest, most intuitive forms.