Problem 64 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(b^{2}\right)^{5}\left(b^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(b^{16}\).

Step 1: Identify the Problem

We need to simplify the expression \((b^{2})^{5}(b^{3})^{2}\) using the product and power rules for exponents.

Step 2: Apply the Power Rule

According to the power rule \((a^m)^n = a^{m imes n}\), apply it to each part: \((b^{2})^{5} = b^{2 imes 5} = b^{10}\) and \((b^{3})^{2} = b^{3 imes 2} = b^{6}\).

Step 3: Use the Product Rule

Now the expression is \(b^{10} imes b^{6}\). By applying the product rule \(a^m imes a^n = a^{m+n}\), we combine the exponents: \(b^{10+6} = b^{16}\).

Step 4: Write the Final Simplified Expression

The expression simplifies to \(b^{16}\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation involving numbers, where a number, called the base, is raised to an exponent or power. This operation is essential for expressing large numbers concisely and performing calculations efficiently.
Understanding exponentiation is crucial because it underpins much of algebra, calculus, and even real-world applications like computing and physics.

• **Base**: The number that is multiplied by itself.
• **Exponent**: Indicates how many times the base is used as a factor.
• **Power**: The complete expression, e.g., in \(a^n\), \(a\) is the base and \(n\) is the exponent.

These components, when combined, allow us to simplify and manipulate expressions quickly. For example, in the expression \((b^2)^5\), \(b\) is the base and \((2 \times 5)\) denotes multiplying \(2\) five times. This means \(b^2\) is itself being raised to another power, making exponentiation a key tool in mathematics to simplify complex expressions.

Mathematical Simplification

Simplification serves to reduce a complex expression into a more understandable or usable form. The purpose of simplifying mathematical expressions is to make them easier to work with, especially during calculation and analysis. Key strategies for simplification include:

• **Applying Exponent Rules**: Use rules like the product and power rules to reduce the number of redundant operations.
• **Combining Like Terms**: This involves gathering and simplifying terms that are alike. In our scenario, different powers of the same base are combined.
• **Manipulating Fractions and Factors**: Often expressions include fractions or factors requiring additional simplification techniques.

In our exercise, simplification is achieved using the power and product rules, transforming \( (b^2)^5(b^3)^2 \) into the much neater \( b^{16} \). This simplification consolidates the expression, making further operations or integration into larger equations more efficient.

Algebraic Expressions

Algebraic expressions are combinations of variables and constants operating under algebraic rules. These expressions form the foundation of algebra, which is utilized in every branch of mathematics and is applicable in real-world problem-solving.Key features include:

• **Variables**: Symbols representing numbers, commonly letters such as \(x\), \(y\), or, in our case, \(b\).
• **Constants**: Specific numbers that are not affected by changes in variable values.
• **Operators**: Include actions like addition, multiplication, and exponentiation.

The expression \((b^2)^5(b^3)^2\) is algebraic because it combines multiplication and exponentiation over the same variable. Simplifying such expressions requires understanding both individual components and the rules governing their interactions.Algebraic manipulation helps reveal these relationships and simplifies computations significantly, supporting both theoretical mathematics and practical applications in engineering, science, and economics.