Question

Complete the statement using \(>\) or \(<\). $$\left(3^{6} \cdot 3^{12}\right) \geq 3^{72}$$

Short answer

\(3^{18} < 3^{72}\)

Step-by-step solution

Identify the exponent rule

Note that the rule of exponents states that for any numbers a, m, and n, we have \(a^{m} \cdot a^{n} = a^{m+n}\).

Apply the exponent rule

Applying this rule to the left hand side of the inequality gives us \(3^{6} \cdot 3^{12} = 3^{6+12}\).

Simplify the left hand side

Simplifying \(6+12\) gives us \(18\), thus the left hand side simplifies to \(3^{18}\).

Compare the simplified expressions

Now we can compare the two sides. We have \(3^{18} ? 3^{72}\). Since 18 is less than 72, we can see that \(3^{18}\) is less than \(3^{72}\).

Key concepts

Inequalities

Inequalities are mathematical expressions that show the relative size or order of two values. They are a cornerstone in algebra and appear extensively across mathematics and its applications. It's essential to understand the notation used in inequalities: the symbols \(<\) means 'less than', and the symbol \(>\) means 'greater than'. When dealing with inequalities of exponential expressions, remember that if the base is greater than one, increasing the exponent results in a larger number. Conversely, with a base less than one, a larger exponent makes the value smaller.

For example, if we consider the base of 3, which is greater than one, any time we increase the exponent, the value of the whole expression rises. Therefore, if we compare \(3^n\) and \(3^m\) and if \(n > m\), then \(3^n > 3^m\). This understanding is crucial when solving exercises that ask you to complete statements using inequality symbols.

Exponential Expressions

Exponential expressions are written with a base and an exponent and represent repeated multiplication. The base \(a\) is the number being multiplied, and the exponent \(n\) tells us how many times to multiply the base by itself. For example, \(2^3\) means \(2 \times 2 \times 2\).

It gets more complicated when expressions involve multiple exponents and bases. Sometimes, expressions can be simplified using exponent rules, which can significantly ease the process of solving problems.

For example, when you see \(3^{6} \times 3^{12}\), you're dealing with a product of exponents with the same base. This is an excellent time to apply one of the exponent rules which states that \(a^m \times a^n = a^{m+n}\). This rule helps combine the two expressions into a single exponential expression, simplifying the calculation and making it easier to handle.

Simplifying Exponents

Simplifying exponents can often turn a complex-looking expression into something far more manageable. It involves using exponent rules to rewrite expressions in a simplified form without changing their values. For example, one of the basic rules states that when you multiply two powers with the same base, you can add the exponents: \(a^m \times a^n = a^{m+n}\).

Applying this rule enables us to simplify \(3^{6} \times 3^{12}\) to \(3^{18}\), merging the two exponents into one. This simplification is invaluable, especially when dealing with inequalities, as it reduces the expressions to a form which can easily be compared. It's always recommended to simplify before attempting to evaluate or solve exponential inequalities.

When comparing exponential expressions as we do with inequalities, the simplification step allows us to make quick and accurate comparisons, as we can see with the simplified expression \(3^{18}\) being clearly less than \(3^{72}\) because the exponents, 18 and 72, can be compared directly once the bases are the same.