Problem 58 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(b^{3}\right)^{6} $$

Step-by-Step Solution

Answer

Using the power rule, \((b^{3})^{6}\) simplifies to \(b^{18}\).

Step 1: Understand the Power Rule for Exponents

The power rule for exponents states that when raising a power to another power, you multiply the exponents. Mathematically, this is written as \((a^m)^n = a^{m \cdot n}\).

Step 2: Identify the Base and Exponents

Identify the base and the exponents in the expression \((b^{3})^{6}\). Here, the base is \(b\), the first exponent is \(3\), and the outer exponent is \(6\).

Step 3: Apply the Power Rule

Use the power rule \((a^m)^n = a^{m \cdot n}\) on the expression \((b^{3})^{6}\). This results in \(b^{3 \cdot 6}\).

Step 4: Multiply the Exponents

Multiply the exponents \(3\) and \(6\). Calculate \(3 \cdot 6 = 18\).

Step 5: Write the Final Expression

Substitute the result from the previous step back into the expression: \(b^{3 \cdot 6} = b^{18}\).

Key Concepts

Simplifying Expressions with Exponents

Simplifying expressions is a key part of learning algebra and involves reducing an expression to its simplest form. When expressions include exponents, we often need special rules to simplify them effectively.
In the current example, we are working with

• Nested exponents: These are exponents that appear within another exponent, like \((b^{3})^{6}\).

To simplify such expressions, the power rule for exponents is your best friend. This rule tells us exactly how to handle the situation when raising an exponent to another exponent. Just remember: multiply the exponents together. By applying this rule step-by-step, you reduce the complexity of the expression significantly and arrive at \(b^{18}\).

• Always identify your base and exponents first.
• Follow mathematical rules like the power rule for consistency.

Understanding Exponents

An exponent is a way of expressing repeated multiplication of a base. For instance, \(a^3\) means you multiply the base \(a\) by itself three times: \(a\times a\times a\).
The whole idea of exponents stems from making complex mathematical operations simpler and more compact. Here are some core points about exponents:

• They appear as small numbers above and to the right of a base.
• They tell you how many times to multiply the base by itself.

When dealing with nested exponents, like in \((b^3)^6\), it becomes crucial to remember specific rules like the power rule so you can simplify correctly without error. The power rule allows you to simplify nested exponents efficiently: multiply the exponents together to get the simplified result, making complex operations manageable.

Using Mathematical Notation

Mathematical notation is like the language of math. It's a system of symbols and signs that express mathematical thoughts concisely and clearly.
When learning mathematics, understanding the notation is crucial:

• It allows for efficient and clear communication.
• It standardizes math so everyone interprets it the same way.
• It simplifies complex math concepts into understandable symbols.

In our example, the expression \((b^3)^6\) is a great illustration of how mathematical notation can compactly represent operations. Each symbol, from the parentheses to the exponents, has a purpose and a rule behind it. The parentheses indicate the nested operation needing simplification. The exponents convey how many times a multiplication occurs. Understanding and reading such notation is fundamental for solving and simplifying algebraic expressions efficiently.