Problem 38 - Algebra 2

Question

Simplify each expression. $$ \left(a^{-\frac{2}{3}}\right)^{-\frac{1}{6}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^{\frac{1}{9}} \).

Step 1: Apply the Power of a Power Rule

The expression \( \left(a^{-\frac{2}{3}}\right)^{-\frac{1}{6}} \) can be simplified using the power of a power rule. According to this rule, if \( (a^m)^n \) is given, it simplifies to \( a^{m \times n} \). So, we multiply the exponents: \[ -\frac{2}{3} \times \left(-\frac{1}{6}\right) \]

Step 2: Multiply the Exponents

Multiply \(-\frac{2}{3}\) by \(-\frac{1}{6}\). When multiplying fractions, multiply the numerators together and the denominators together: \[ -\frac{2}{3} \times \left(-\frac{1}{6}\right) = \frac{2 \times 1}{3 \times 6} = \frac{2}{18} \] This simplifies to \( \frac{1}{9} \) by dividing the numerator and the denominator by 2.

Step 3: Write the Simplified Expression

Now, replace the original expression \( \left(a^{-\frac{2}{3}}\right)^{-\frac{1}{6}} \) with the simplified exponent obtained, \( a^{\frac{1}{9}} \). Therefore, the simplified form of the given expression is \( a^{\frac{1}{9}} \).

Key Concepts

Power of a Power Rule

When we deal with expressions that involve exponents raised to another exponent, we use the **Power of a Power Rule**. This rule is like stacking powers on top of each other. If you have an expression like \((a^m)^n\), the power of a power rule allows you to simplify it by multiplying the exponents together. This results in a new expression, \(a^{m \times n}\).
For example, if you have \((x^2)^3\), you apply the rule and get \(x^{2 \times 3} = x^6\). In the same way, in our original exercise, we started with \( \left(a^{-\frac{2}{3}}\right)^{-\frac{1}{6}} \). Using the power of a power rule here, we multiply the exponents

• \(-\frac{2}{3} \times -\frac{1}{6}\)

resulting in a new exponent value that simplifies the whole expression into \(a^{\frac{1}{9}}\).
This rule is super handy and saves a lot of time when simplifying complex expressions. Remember, the base "a" stays the same, and only the exponents are affected.

Simplifying Expressions

Simplifying expressions makes them neater and often easier to work with. The ultimate goal is to condense them into the simplest form possible. Simplifying involves reducing fractions, combining like terms, and eliminating negative signs when appropriate.
In the exercise we discussed, the simplifying process involved working with fractions in the exponents. We first used the power of a power rule to combine the exponents. Then, we performed an operation—multiplying fractions—to find a single simplified exponent. Simplified expressions are the ones with the least number of terms while maintaining equivalent mathematical value.
The final step in our example led us to a more manageable expression, \(a^{\frac{1}{9}}\), which is easier to understand and use in further calculations or substitutions. Always strive to simplify an expression step by step, ensuring all rules and operations are applied correctly. This will uphold both the expression's integrity and its mathematical meaning.

Multiplying Fractions

Multiplying fractions is often considered easier than adding or subtracting them. When multiplying fractions, you simply multiply the numerators (the numbers on top) together and the denominators (the numbers on the bottom) together. For instance, suppose you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\). To multiply them, you calculate:

• \(\frac{a \times c}{b \times d}\)

In our example exercise, we had to multiply \(-\frac{2}{3}\) and \(-\frac{1}{6}\). Here's how it works:
First, multiply the numerators: \(-2 \times -1 = 2 \).
Then multiply the denominators: \(3 \times 6 = 18\).
This gives us the result: \(\frac{2}{18}\). Often, the next step involves simplifying the resulting fraction. In this case, \(\frac{2}{18}\) simplifies to \(\frac{1}{9}\).
Recognizing how to break down and simplify fractions is essential when dealing with expressions, especially when exponents are involved in multiplying fractions.