Problem 16 - Elementary Algebra

Question

For the following problems, expand each product (do not compute the actual value). \(2^{7} \cdot 3^{4}\)

Step-by-Step Solution

Answer

Question: Expand the product \(2^7 \cdot 3^4\) without computing its actual value. Answer: \(2^7 \cdot 3^4 = 2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot2\cdot2\)

Step 1: Rewrite the given product in terms of repeated multiplications

The given product is \(2^7 \cdot 3^4\). We will rewrite it as the product of the repeated factors (2 and 3) by expanding the exponentiation: \(2^7 \cdot 3^4 = (2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2) \cdot (3\cdot3\cdot3\cdot3)\)

Step 2: Rearrange the factors

Now, we will rearrange the factors in a way that makes the expansion clearer: \((2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2) \cdot (3\cdot3\cdot3\cdot3) = 2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot2\cdot2\) This rearranged product has the same factors as our original expression, but they have been reordered to better visualize the expansion.

Step 3: Write the final expanded expression

Finally, write down the expanded expression as a product of the individual factors: \(2^7 \cdot 3^4 = 2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot3\cdot2\cdot2\cdot2\) This is the expanded product without computing its actual value.

Key Concepts

Exponential Expressions

Exponential expressions are a key concept in mathematics and are frequently used in various areas such as algebra and calculus. An exponential expression is typically written in the form of a base raised to an exponent or power, like \(a^n\). Here, \(a\) is the base, and \(n\) is the exponent. Exponents indicate how many times the base is multiplied by itself. For example, in the expression \(2^7\), we have 2 multiplied by itself 7 times. This can be expanded as:

• \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)

Recognizing these components can make it easier to manipulate and solve equations involving exponents, even without calculating the exact product of these numbers. This basic understanding is crucial in mathematics education and critical thinking.

Expansion of Products

Expanding products involves breaking down expressions with multiplication into their individual factors, which often include exponential expressions. This arithmetic skill is helpful for simplifying complex equations and enhancing problem-solving abilities.When faced with a product like \(2^7 \cdot 3^4\), rather than multiplying the bases immediately, we can expand each term separately as:

• \(2^7 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)
• \(3^4 = 3 \cdot 3 \cdot 3 \cdot 3\)

Combining these expansions allows us to rearrange and better visualize the expression:

• \(2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 2 \cdot 2\)

Reordering doesn't affect the outcome but provides a clearer picture of the entire multiplication process. This clarity is significant for students, facilitating comprehension and mastery over mathematical expressions.

Mathematics Education

Mathematics education benefits greatly from a simplified explanation of complex ideas, such as exponential expressions and the expansion of products. The goal is to foster a deeper understanding among students and to cultivate the ability to apply these concepts to solving practical problems. Teaching tools, like expanding exponential expressions, enable students to visualize and understand multiplication in a broader context. Instructors emphasize the importance of breaking down problems step by step to avoid overwhelming learners:

• Step 1: Identify the base and exponent.
• Step 2: Expand each exponential term into repeated multiplication.
• Step 3: Rearrange and merge factors for a better understanding.

This structured approach nurtures confidence in mathematical capabilities and promotes analytical skills. Incorporating these strategies into the curriculum ensures students build a solid foundation in math, preparing them for more advanced concepts as they progress in their education journey.