Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(b^{-3} c\right)^{3}\)

Step-by-Step Solution

Answer

\(\frac{c^3}{b^9}\)

Step 1: Apply the Power of a Power Rule

The given expression is \( \left(b^{-3} c\right)^{3} \). Apply the power of a power rule \( (x^a)^b = x^{a \cdot b} \) to each term inside the parenthesis: \((b^{-3})^3\) and \(c^3\).

Step 2: Simplify Each Term

For the term \((b^{-3})^3\), using the power of a power rule, we have \(b^{-3 \times 3} = b^{-9}\). For the term \((c)^3\), it simplifies to \(c^3\).

Step 3: Combine the Simplified Terms

Combine the simplified terms: \(b^{-9} \cdot c^3\).

Step 4: Convert to Positive Exponents

Since we need to express the answer with positive exponents, convert \(b^{-9}\) to \(\frac{1}{b^9}\). The expression becomes \(\frac{c^3}{b^9}\).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is a fundamental concept in exponent rules. This rule is used when you have an exponent raised to another exponent, which is represented as \((x^a)^b\). According to this rule, you can simplify it by multiplying the exponents together. So, you get \(x^{a \cdot b}\).
For example, let's consider the expression \((b^{-3})^3\). By applying the power of a power rule, you multiply the exponents:

• Inside the parenthesis, the base is \(b\) with an exponent of \(-3\).
• The outer exponent is \(3\).
• Applying the rule: \((-3) \times 3 = -9\). Hence, it simplifies to \(b^{-9}\).

Always remember, even with negative exponents, the rule holds.
This understanding is essential for simplifying complex exponential expressions effectively.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form while maintaining their value. This process makes them easier to work with in mathematical operations. One way to simplify expressions is by applying exponent rules, such as the Power of a Power Rule.
In the expression \(\left(b^{-3} c\right)^3\):

• Apply the Power of a Power Rule to each term inside, resulting in \(b^{-9}\) and \(c^3\).
• The simplified expression becomes \(b^{-9} \cdot c^3\).

This expression is simplified yet still contains a negative exponent, which is addressed next. Being able to identify and correctly apply these rules helps in simplifying expressions systematically.

Positive Exponents

Positive exponents mean expressing numbers in a format where the exponent is greater than zero. This is preferred as it makes expressions clearer and more useful in calculators and mathematical software.
When you encounter a negative exponent, such as \(b^{-9}\), you can rewrite it using positive exponents like this:

• Recall that \(x^{-a} = \frac{1}{x^a}\).
• So, \(b^{-9}\) becomes \(\frac{1}{b^9}\).

This conversion helps in expressing the answer in positive form, which is often required in final answers.
Finally, the expression \(b^{-9} \cdot c^3\) is transformed to \(\frac{c^3}{b^9}\), ensuring that all exponents are positive. This step completes the simplification process, making the expression ready for further mathematical operations.