Problem 63 - Algebra 1

Question

Complete the statement using \(>\) or \(<\). $$(5 \cdot 6)^{4} \underline{?} 5 \cdot 6^{4}$$

Step-by-Step Solution

Answer

\((5 \cdot 6)^{4} > 5 \cdot 6^{4}\)

Step 1: Calculate the first expression

In the expression \((5 \cdot 6)^{4}\), the operation within the parentheses should be done first. So, firstly, \(5 \cdot 6\) is calculated which equals \(30\). Now, calculate \((30)^4\) , which equals \(810,000\).

Step 2: Calculate the second expression

In the expression \(5 \cdot 6^{4}\), according to the order of operation, exponentiation should be done first before multiplication. Therefore, first calculate \(6^{4}\) which equals \(1296\). Then, multiply this result by \(5\), so \(5 \cdot 1296\) equals \(6480\).

Step 3: Compare the values

Now that both calculations have been made, their results can be compared. \(810,000\) is greater than \(6480\), so the correct symbol to use is \(>\).

Step 4: Final Statement

Therefore, the complete statement would be \((5 \cdot 6)^{4} > 5 \cdot 6^{4}\).

Key Concepts

Exponents

Exponents play a crucial role in mathematics by showing how many times a number, referred to as the base, is multiplied by itself. Understanding exponents is essential for performing calculations accurately and efficiently.

When dealing with expressions that involve exponents, the order in which you perform calculations is important as it affects the final result. Take the expression - - In - For example, in - The parentheses indicate that multiplication within We calculate \((5 \cdot 6)\) first to get \(30\). This makes our expression - Thereafter, \(30\) is raised to the power of 4, denoting \(30\) multiplied by itself 3 more times, resulting in \(810,000\).

This exemplifies how different placements of parentheses can significantly alter the outcome of an expression involving exponents.

Multiplication

Multiplication is a fundamental operation in mathematics, used for calculating products of numbers. It involves repeated addition of a number as many times as the value of another number — for example, multiplying 5 by 6 means adding 5 together 6 times.

When multiplication is combined with other operations like addition or subtraction, it is crucial to follow the correct order of operations—commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In the given problem, both expressions involve multiplication, but placed differently:

• \((5 \cdot 6)^{4}\) - multiplication inside the parentheses first, leading to a higher power
• \(5 \cdot 6^{4}\) - exponentiation of 6 before multiplying by 5, a different approach altogether leading to a smaller product

This difference showcases how multiplication interacts uniquely with exponents and parenthetical grouping.

Comparing Numbers

Comparing numbers is an important mathematical skill that involves determining if a number is larger, smaller, or equal to another. It uses symbols like \(>\), \(<\), and \(=\).

In comparing complex expressions like \((5 \cdot 6)^{4}\) and \(5 \cdot 6^{4}\), breaking them down into simpler calculations helps:

• Calculate \((5 \cdot 6)^{4} = 810,000\)
• Calculate \(5 \cdot 6^{4} = 6480\)

After solving, compare the results directly:

• Since \(810,000\) is much greater than \(6480\), the symbol to use is \(>\)

This method ensures precise comparisons and clearer understanding of value relationships.