Problem 19 - Beginning and Intermediate Algebra

Question

Simplify. $$3^{-3} \cdot 3^{8} \cdot 3$$

Step-by-Step Solution

Answer

The simplified expression of \(3^{-3} \cdot 3^{8} \cdot 3\) is 729.

Step 1: Understand the properties of exponents

Let's recall the properties of exponents which will help us in simplifying the expression. Specifically, we will use the product of powers property: \(a^{m} \cdot a^{n} = a^{m+n}\), where a is the base and m and n are the exponents.

Step 2: Apply the product of powers property

Now, apply the product of powers property to the given expression: \(3^{-3} \cdot 3^{8} \cdot 3\) Since all the base numbers are 3, we can add their exponents: \(-3+8+1\)

Step 3: Add the exponents

Add the exponents as follows: \(-3+8+1 = 6\)

Step 4: Substitute the sum of the exponents

Now that we have the sum of the exponents, we just need to substitute it back into the expression: \(3^{6}\)

Step 5: Calculate the simplified expression

Now, calculate the simplified expression: \(3^{6} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729\) So, the simplified expression is 729.

Key Concepts

Simplifying Expressions

When we are faced with a mathematical expression, simplifying it can help us understand it better. Simplification often means reducing the expression to its most basic form.
It's like cleaning up to make things easier to manage or understand.In the context of expressions with exponents, simplification involves using specific rules to condense multiple terms into one. For example, in the given exercise, we're simplifying the expression \(3^{-3} \cdot 3^{8} \cdot 3\) into a single term with a single exponent. This is achieved by using rules about exponents that allow us to combine like terms efficiently.

By simplifying expressions:

• We make calculations more straightforward.
• We can easily compare and evaluate expressions.
• We can more quickly see the underlying structure of a problem.

Remember, the goal of simplifying is to make life easier, so always look for ways to apply these techniques.

Product of Powers

The product of powers property is a cornerstone of working with exponents. It states that when you multiply powers with the same base, you can add their exponents: \(a^{m} \cdot a^{n} = a^{m+n}\).
This property helps to quickly combine terms that share the same base, which is crucial for simplifying expressions.In the exercise, each component of the expression \(3^{-3} \cdot 3^{8} \cdot 3\) had the same base: '3'. So, instead of handling them separately, we utilized the product of powers property to merge them into a single term by adding the exponents \(-3\), \(8\), and the implicit \(1\) that accompanies \(3\) (since \(3\) is the same as \(3^1\)).

This results in a single expression with an exponent of \(6\). Using the product of powers property saves time and simplifies the solving process. Always remember:

• This property is only applicable when the bases are the same.
• Remember to consider any implicit exponents (e.g., an unwritten \(1\)).
• Combine the exponents by simple addition.

Mastering this property enables you to tackle a wide array of problems involving exponents.

Exponent Rules

Exponent rules are the building blocks for handling exponential expressions. They include several key principles for operations involving powers, each aiding in the simplification and computation process.
These rules not only apply to positive exponents but also to zero and negative ones.For the original exercise, we made good use of:

• Product of Powers: As we've discussed, this rule allows us to add the exponents when multiplying like bases.
• Negative Exponent Rule: This tells us that a negative exponent means division by that power: \(a^{-m} = \frac{1}{a^m}\). It helps convert negative exponents into fractions if needed.

Here, the negative exponent \(-3\) indicated division by \(3^3\), but since it turned into part of a larger product, it was handled through addition.

Other important exponent rules include:

• Zero Exponent Rule: Any non-zero base with an exponent of zero is \(1\), i.e., \(a^0 = 1\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^{m})^{n} = a^{m \cdot n}\).
• Quotient of Powers: Divide same bases by subtracting the exponents: \(a^{m} / a^{n} = a^{m-n}\).

Understanding these rules will greatly aid in both simplifying expressions and in performing more complex calculations. Practice applying them, and soon they will be second nature!