Question

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

Short answer

The correct relation between the two expressions is \( \left(6^{2} \cdot 3\right)^{3} > 6^{5} \cdot 3^{3} \)

Step-by-step solution

Simplify the left expression

Using the property of powers we can simplify the left side of the inequality: \( \left(6^{2} \cdot 3\right)^{3} = (6^{2})^{3} \cdot 3^{3} = 6^{6} \cdot 3^{3} \)

Simplify the right expression

The right side of the inequality is already simplified and stays as: \( 6^{5} \cdot 3^{3} \)

Compare the expressions

Now to determine which sign we should use in the expression \( 6^{6} \cdot 3^{3} \_ 6^{5} \cdot 3^{3} \), we see that both expressions have \( 3^{3} \). Moreover, the term with 6 on the left side has a higher exponent than on the right side, i.e. power of 6 is greater in the left side expression. So, the left side of the expression will definitely be greater than the right one. Hence, the right sign here is '>'. Hence, \( \left(6^{2} \cdot 3\right)^{3} > 6^{5} \cdot 3^{3} \)

Key concepts

Exponent Rules

Understanding exponent rules is essential when dealing with algebraic expressions, especially when working with inequalities in algebra. Exponent rules include operations such as multiplication, division, and raising powers to another power. When you multiply two powers with the same base, you add the exponents, as seen in the expression \(a^m \cdot a^n = a^{m+n}\). Conversely, when dividing two powers with the same base, you subtract the exponents \(\frac{a^m}{a^n} = a^{m-n}\). Another important rule is when you raise a power to another power, you multiply the exponents. This is shown by \( (a^{m})^{n} = a^{m \cdot n} \).
In the given exercise example, the property \( (a^{m})^{n} = a^{m \cdot n} \) was used to simplify the left expression from \( (6^{2} \cdot 3)^{3} \) to \(6^{6} \cdot 3^{3}\). This transformation directly leads to the understanding that the expression with the larger exponent \(6^{6}\) is greater than \(6^{5}\), thus confirming that the correct inequality sign to use is '>.'

Properties of Powers

The properties of powers play a significant role in simplifying algebraic expressions involving exponents. Some of the key properties include the power of a product and the power of a power, which were utilized in solving the given inequality. The power of a product property indicates that when an entire product is raised to an exponent, you raise each factor to that same exponent, as in \( (ab)^{n} = a^{n}b^{n} \).

Power of a Power

Similarly, when you have a power raised to another power, you multiply the exponents, reinforcing the idea that \( (a^{m})^{n} = a^{m \cdot n} \).
These properties enabled the simplification from the original expression \( (6^{2} \cdot 3)^{3} \) to \(6^{6} \cdot 3^{3}\), highlighting how understanding the properties of powers allows for efficient comparison of algebraic expressions involving exponents. These properties are fundamental to performing operations correctly and reaching the proper inequality conclusion.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They become particularly interesting when they contain exponents or variables raised to powers. To work with algebraic expressions effectively, it's important to be familiar with techniques such as expanding, factoring, and simplifying expressions.
In the exercise, we deal with an algebraic expression that is part of an inequality. While the expressions on both sides of the inequality have similar bases and factors, their exponents differ, which significantly affects their values. Through simplifying expressions using exponent rules and properties of powers, we can evaluate and compare the sizes of different algebraic expressions as seen in this exercise.
Always remember that simplifying algebraic expressions is not just about applying the rules mechanically, but also about understanding the underlying concepts. This understanding ensures that you can navigate through the complexities of algebra with greater ease and accuracy, and ultimately determine the correct relationship between expressions as required by an inequality.