Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(b^{3} c^{4}\right)^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \( b^6 c^8 \).

Step 1: Apply Power Rule

The first step is to apply the Power Rule of exponents, which states that \( (a^m)^n = a^{m imes n} \) for any base \( a \) and exponents \( m \) and \( n \). Here, we need to distribute the outer exponent of 2 to both \((b^{3})\) and \((c^{4})\).

Step 2: Simplify Each Term

Simplify each part of the expression by multiplying the exponents. For \( b^{3} \), we have \( (b^3)^2 \), which becomes \( b^{3 imes 2} = b^6 \). Similarly, for \( c^{4} \), we have \( (c^4)^2 \), which becomes \( c^{4 imes 2} = c^8 \).

Step 3: Write the Final Expression

Combine the simplified terms from the previous step to give the final expression: \( b^6 c^8 \). This expression has all the exponents simplified and is expressed using positive exponents.

Key Concepts

Power Rule

The power rule is a fundamental concept in the world of exponents. It helps simplify expressions that involve raising a power to another power. The rule can be expressed as: \[(a^m)^n = a^{m \times n}\]Here,

• \(a\) represents the base,
• \(m\) is the initial exponent,
• and \(n\) is the power to which the base-exponent is raised.

To use the power rule, you simply multiply the exponents together. For instance, in the expression \((b^3 c^4)^2\), we distribute the outer exponent of 2 to each component inside the parentheses. This involves applying the power rule to both \(b^3\) and \(c^4\), leading to \(b^{3 \times 2}\) and \(c^{4 \times 2}\) respectively. This simplifies to \(b^6\) and \(c^8\). Mastering the power rule is crucial for simplifying more complex expressions with nested exponents.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with by reducing them to their simplest form. This process involves using different algebraic rules and operations, including the power rule. When simplifying expressions with exponents, the goal is to have each base with a single positive exponent.In our example expression \((b^3 c^4)^2\), after applying the power rule, we get \(b^6\) and \(c^8\). These are ready to be combined into a single expression because they have been reduced to their simplest exponents.Remember to:

• Always multiply exponents when applying the power rule.
• Combine like terms when possible.
• Ensure all final exponents are positive.

By consistently following these steps, the process of simplifying complex expressions becomes straightforward and systematic.

Positive Exponents

Expressing answers with positive exponents is essential in both academic and real-world contexts, as it simplifies further mathematical operations. Negative exponents can make expressions harder to interpret and work with.A positive exponent indicates how many times a base is multiplied by itself. For example, in \(b^6\), the base \(b\) is multiplied by itself six times:\[b \times b \times b \times b \times b \times b\]In contrast, a negative exponent represents the reciprocal of the positive exponent. Hence, \(b^{-6}\) would mean \(\frac{1}{b^6}\). But in most cases, especially in simplified answers, positive exponents are preferred.When simplifying expressions, always check that all exponents are positive. This practice not only helps make the expressions clearer but also prepares them for further simplification or calculation.