Problem 49 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(a^{-5} b^{2}\right)^{-6} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{a^{30}}{b^{12}}\).

Step 1: Apply the Power of a Power Property

When we have an exponent raised to another exponent, we multiply the exponents together. We start with \((a^{-5} b^{2})^{-6}\). Applying the power of a power property: \[(a^{-5})^{-6} ext{ and } (b^{2})^{-6}\] This results in: \[a^{-5 imes -6} = a^{30} ext{ and } b^{2 imes -6} = b^{-12}\]

Step 2: Simplify the Expression

Now that we have applied the exponents, our expression \((a^{-5})^{-6} (b^{2})^{-6}\) becomes:\[a^{30} b^{-12}\]

Step 3: Convert Negative Exponents to Positive

To write each part of the expression with positive exponents, use the rule that any base with a negative exponent can be written as the reciprocal with a positive exponent. Therefore, the term \(b^{-12}\) can be rewritten as \(\frac{1}{b^{12}}\).Thus, the expression \(a^{30} b^{-12}\) becomes:\[a^{30} \cdot \frac{1}{b^{12}} = \frac{a^{30}}{b^{12}}\]

Key Concepts

Power of a Power Property

When you see an exponent raised to another exponent, use the power of a power property. This rule simplifies expressions by multiplying the exponents together. For example, given the expression \((x^m)^n\), the property tells us to calculate \(x^{m \times n}\). In our problem, the expression \((a^{-5} b^{2})^{-6}\) requires this property to simplify it correctly.

• For \((a^{-5})^{-6}\), we calculate \(a^{-5 \times -6}\), which simplifies to \(a^{30}\).
• Similarly, for \((b^2)^{-6}\), we find \(b^{2 \times -6}\), which becomes \(b^{-12}\).

Now, the problem is reduced to managing these simplified expressions.

Negative Exponents

Understanding negative exponents is crucial when simplifying expressions. A negative exponent on a number or variable signifies a reciprocal. This means that \(x^{-n}\) can be written as \(\frac{1}{x^n}\). Consider the expression involving \(b^{-12}\): we follow the rule of turning the negative exponent into a positive one by finding its reciprocal:

• The expression \(b^{-12}\) translates to \(\frac{1}{b^{12}}\).

This transformation allows us to work effectively towards expressing everything with positive exponents.

Positive Exponents

After applying rules like the power of a power property and converting negative exponents, positive exponents make calculations easier. They represent straightforward multiplication of a base by itself. For instance, in the expression \(a^{30}\), it means we multiply \(a\) by itself 30 times. Working with positive exponents involves:

• Executing operations with bases easily and accurately.
• Ensuring results are expressed clearly and simplified where possible.

Maintaining positive exponents in final results ensures clarity and uniformity in mathematics.

Reciprocals

Reciprocals help convert expressions with negative exponents into their positive counterparts. The reciprocal of a number is one divided by that number.

• For \(x^{-n}\), the reciprocal is \(\frac{1}{x^n}\).

This essential math concept simplifies expressions and makes them easier to interpret. For example, let's look at our expression \(b^{-12}\). By taking the reciprocal:

• \(b^{-12}\) becomes \(\frac{1}{b^{12}}\), effectively eliminating the negative exponent.

Using reciprocals ensures that results maintain a standard form, often required in math and science.