Problem 45 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(\frac{a^{2}}{4}\right)^{3} $$

Step-by-Step Solution

Answer

\(\frac{a^6}{64}\).

Step 1: Understand Exponent Rules

When raising a power to another power, you multiply the exponents. So, if you have \((a^m)^n\), it becomes \(a^{m\cdot n}\). This rule will help simplify \(\left(\frac{a^2}{4}\right)^3\).

Step 2: Apply the Exponent Rule to the Numerator

Focus on the numerator of the fraction first. We have \((a^2)^3\), which according to the power of a power rule is \(a^{2 \cdot 3} = a^6\).

Step 3: Apply the Exponent Rule to the Denominator

The denominator is \(4\), which is equivalent to \((2^2)^3\). Applying the power of a power rule, this becomes \(2^{2 \cdot 3} = 2^6\).

Step 4: Combine the Simplified Numerator and Denominator

Now combine both the numerator and the denominator to get the simplified expression: \(\frac{a^6}{2^6}\).

Step 5: Simplify Further if Possible

The expression cannot be simplified further in terms of the base or the exponent, so the result remains \(\frac{a^6}{64}\) since \(2^6 = 64\).

Key Concepts

Power of a Power Rule

When dealing with expressions like \((a^m)^n\), we encounter the Power of a Power Rule. This rule is pivotal because it tells us how to simplify complex expressions involving stacked exponents. The rule states that when you raise a power to another power, you simply multiply the exponents. For instance, \((a^m)^n\) simplifies to \(a^{m \cdot n}\).
This means, instead of potentially expanding an expression repeatedly, you can quickly find the result by calculating directly. This technique saves time and avoids errors.

• Example: For \((x^3)^4\), apply the rule to get \(x^{3 \cdot 4} = x^{12}\).
• Application: Use this rule whenever you see a power raised over another power in both numerators and denominators.

Numerator and Denominator Simplification

Simplifying the numerator and denominator individually helps to make complex expressions more manageable. Always simplify each part separately before combining them.

In our exercise, for the numerator \((a^2)^3\), applying the power of a power rule results in \(a^{2 \cdot 3} = a^6\). This simplification is straightforward given the rule explained earlier.

• Denominator: The denominator, initially represented as \(4\), needs conversion into an expression suitable for exponentiation. Rewrite it as \((2^2)^3\), applying the same rule yields \(2^{2 \cdot 3} = 2^6\), and simplifies to 64.
• Combine Both Parts: Once simplified separately, you can combine them to form the simplified fraction \(\frac{a^6}{64}\).

Fractional Exponents

Fractional exponents are another way to express roots and powers elegantly, combining both concepts into a single notation. When you have expressions like \(a^{m/n}\), it translates to either a root or an exponent form.

The numerator 'm' of the fractional exponent indicates the power, while the denominator 'n' signifies the root. So \(a^{1/2}\) matches with \(\sqrt{a}\), and \(a^{3/2}\) translates to \(a^3\) under a square root. This notation is very powerful for simplifying and working with root expressions.

• Understanding: Think of a fractional exponent as two expressions in one: a root (root part) and a power (exponent part).
• Versatility: Fractional exponents allow smooth transition between root and power computations, making them invaluable in calculus and algebra.