Problem 95 - Algebra for College Students

Question

Your friend simplifies \(2^{3} \cdot 2^{2}\) as follows: $$ 2^{3} \cdot 2^{2}=4^{3+2}=4^{5}=1024 $$ What has she done incorrectly and how would you help her?

Step-by-Step Solution

Answer

The mistake was adding exponents of different bases. Correctly, \(2^3 \cdot 2^2 = 2^5 = 32\).

Step 1: Identify the Problem

The mistake in your friend's solution is in the application of exponent rules. She incorrectly added the exponents of different bases, treating the bases as the same. Instead, both bases should remain 2.

Step 2: Recognize the Rule for Same Bases

When you multiply powers with the same base, you keep the base the same and add the exponents: \( a^m \cdot a^n = a^{m+n} \).

Step 3: Apply the Correct Rule

Using the correct rule for the expression \(2^3 \cdot 2^2\), add the exponents: \( 3 + 2 = 5 \), so \(2^3 \cdot 2^2 = 2^{3+2} = 2^5\).

Step 4: Calculate the Correct Answer

Now compute \(2^5\): Multiply 2 by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

Key Concepts

Multiplication of Powers

When dealing with exponentiation, one important concept is **multiplication of powers**. It involves multiplying expressions that both have exponents. The first key thing to remember is that the base numbers must be the same for the multiplication to simplify according to the exponent rules. For example, with expressions like \(2^3 \cdot 2^2\), the bases, "2," are the same. This allows us to proceed with multiplication.

So, what do we do with the exponents? For identical base numbers, we add the exponents. Thus, using the rule: \(a^m \cdot a^n = a^{m+n}\), we apply it as follows:

• Keep the base "\(a\)" unchanged.
• Add the exponents "\(m\)" and "\(n\)" together.
• Write the new expression with the summed exponents.

Using the initial example \(2^3 \cdot 2^2\), this results in \(2^{3+2} = 2^5\). Remember, this rule only applies when the bases are identical.

Same Base Exponentiation

**Same base exponentiation** talks about situations where two exponential expressions share the same base. This concept is crucial because it represents a specific class of problems where we can simplify more efficiently. While handling these cases:

Focus on maintaining the base constant. The similarity in the base allows us to streamline calculations, making it simpler to handle powers. The process always involves adding the exponents together, regardless of their initial values.

Let's say we have \(b^4 \cdot b^2\):

• The base "\(b\)" stays unchanged.
• Add the exponents: \(4 + 2 = 6\).
• Conclude with \(b^{4+2} = b^6\).

Remember, due to the rules of mathematics, the integrity of the base is essential. The method only works if you aren't changing or mixing different bases.

Algebraic Mistakes

**Algebraic mistakes** often occur when working with expressions, especially when dealing with exponents. It's easy to make errors if the rules of exponents aren't applied correctly. A very common mistake students make, as seen in the example, arises from improperly handling bases and exponents.

One such error is treating different bases as if they were the same. For instance, confusing \(2^3 \cdot 2^2\) by incorrectly changing and adding the bases, turning it into something like \(4^{3+2}=4^5\). This shows a misunderstanding of the exponent rules, leading to the wrong solution.

To avoid such errors, keep these in mind:

• Check your bases - ensure they match before adding exponents.
• Follow exponent rules diligently without assuming shortcuts.
• Always verify calculations by breaking down the steps.

By staying vigilant and practicing frequently, you can identify and rectify algebraic mistakes effectively.