Problem 64 - Algebra 1

Question

Complete the statement using \(>\) or \(<\). $$5^{2} \cdot 3^{6} \underline{?}(5 \cdot 3)^{6}$$

Step-by-Step Solution

Answer

\(5^{2} \cdot 3^{6} < (5 \cdot 3)^{6}\)

Step 1: Break down the expressions

Examine both expressions: \(5^{2} \cdot 3^{6}\) and \((5 \cdot 3)^{6}\). If we calculate the first, we get \(25 \cdot 729 = 18225\). The second one is \((15)^{6}\), which simplifies to a big number, \(11390625\). It's clear that \(18225 < 11390625\).

Step 2: Formulate the final statement

Based on the calculations, we can insert the inequality sign in the initial mathematical statement, which results in \(5^{2} \cdot 3^{6} < (5 \cdot 3)^{6}\).

Key Concepts

Exponentiation

When we talk about exponentiation in algebra, we are referring to the process of raising a number, called the base, to a certain power. The power, or exponent, tells us how many times to multiply the base by itself. For example, in the expression \(5^{2}\), 5 is the base and 2 is the exponent, which tells us that we need to multiply 5 by itself once: \(5 \times 5 = 25\).

It is crucial to recognize that exponentiation is not the same as multiplication, even though they are related operations. Multiplying a number by itself a certain number of times is what exponentiation essentially captures. A common mistake is to multiply the base by the exponent, which is not correct. Instead, you should understand that \(5^{2}\) is not \(5 \times 2\) but is actually \(5 \times 5\).

An important property of exponentiation is that when a product of bases is raised to an exponent, each base is raised to the power separately. For example, \((5 \cdot 3)^{6}\) means that both 5 and 3 are being raised to the 6th power, and then multiplied together: \(5^{6} \cdot 3^{6}\). This is different from \(5^{2} \cdot 3^{6}\), as in our original exercise.

Comparing Numbers

The skill of comparing numbers is essential in mathematics, especially while dealing with inequalities where one number is either greater than or less than another. When comparing numbers with large values, such as in the case of exponentiation, it can get particularly tricky. In the provided exercise, we compared \(5^{2} \cdot 3^{6}\) with \((5 \cdot 3)^{6}\), and it was evident after calculation that the second number is much larger.

A key strategy in comparing large numbers is to look for properties that can simplify the process. For instance, knowing that exponentiation has precedence over multiplication can help us decide which number is greater without calculating the full value. Understanding and applying these properties correctly can save time and avoid potential errors. Always remember to keep your work organized, and compare step by step when you handle expressions with different operations and large numbers.

Order of Operations

The order of operations is a set of rules that mathematicians agree upon to avoid confusion when interpreting and solving mathematical expressions. In our exercise, we used the order of operations to simplify and compare the two numbers. The order of operations can be remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given expressions, exponentiation comes before multiplication. This rule is why \((5 \cdot 3)^{6}\) simplifies to \(15^{6}\), and we perform the exponentiation before considering the multiplication. Ignoring this rule could lead to the incorrect assumption that the two expressions \(5^{2} \cdot 3^{6}\) and \((5 \cdot 3)^{6}\) are equal, which is not the case. This highlights the importance of following the order of operations for accurate calculations and comparisons.