Problem 31 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(\left(2^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The expression \(\left(2^{3}\right)^{2}\) simplified as a single power of the base is \(2^{6}\).

Step 1: Identify the base and the powers

The given expression is \(\left(2^{3}\right)^{2}\). From the expression, the base is '2' and the powers are '3' and '2'.

Step 2: Use the power of a power rule

Apply the power of a power rule, i.e, \((a^{m})^{n} = a^{m \cdot n}\). Here, \(a = 2\), \(m = 3\), and \(n = 2\). This gives \(2^{m \cdot n} = 2^{3 \cdot 2}\).

Step 3: Calculate the exponent

Multiply the two powers together to get the new power. This gives \(2^{3 \cdot 2} = 2^{6}\).

Key Concepts

Power of a Power Rule

When you're dealing with exponents and you see an expression like \( (a^{m})^{n} \), you can use the power of a power rule to simplify it. This rule helps you combine the exponents into a single power. To use this rule, you simply multiply the outer exponent by the inner exponent. In general, this rule is written as:

• \( (a^{m})^{n} = a^{m \cdot n} \)

This makes it much easier to work with the expression. Consider the expression \( (2^{3})^{2} \). Here, the base is 2, and you have an inner exponent of 3 and an outer exponent of 2. By applying the power of a power rule, you multiply 3 and 2, giving you \( 2^{6} \). Using this rule reduces complexity and makes calculations faster and more straightforward.

Base and Exponent

Understanding the concepts of base and exponent is fundamental when working with powers. In the expression \( a^{n} \), the letter \( a \) is known as the base. It represents the number being multiplied. The number \( n \) is the exponent, which tells you how many times to multiply the base by itself. For example, in \( 2^{3} \), 2 is the base and 3 is the exponent, meaning you multiply 2 by itself 3 times. So, \( 2^{3} = 2 \times 2 \times 2 = 8 \).It’s essential to differentiate between the base and the exponent, especially when simplifying or calculating expressions. Recognizing each part helps in applying the appropriate rules, like the power of a power rule, to simplify complex expressions.

Simplifying Expressions

Simplifying expressions involving exponents is all about following certain rules to make expressions easier to handle. The goal is often to express something in a simpler or more compact form without changing its value.When simplifying, start by identifying what you have—like the base and exponents—and what rules apply. Common rules include the power of a power rule, product of powers rule \( a^{m} \cdot a^{n} = a^{m+n} \), and the quotient of powers rule \( \frac{a^{m}}{a^{n}} = a^{m-n} \) when \( m>n \). In the case of \( (2^{3})^{2} \), apply the power of a power rule by multiplying the exponents, resulting in \( 2^{6} \). By simplifying the expression to \( 2^{6} \), you can see that the form becomes much easier to understand. If needed, further simplify by computing \( 2^{6} = 64 \). Simplification makes expressions manageable and prepares them for more advanced mathematical operations.