Problem 70 - Algebra 1: Concepts and Skills

Question

In Exercises \(65-70,\) simplify. Then use a calculator to evaluate the expression. $$ \left(3^{2} \cdot 2^{3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified and calculated expression is \(373248\).

Step 1: Simplifying the expression inside the parentheses

First, we simplify \(3^{2} \cdot 2^{3}\). To do this, calculate both \(3^{2}\) and \(2^{3}\) which will give \(9\) and \(8\) respectively. Now multiply \(9 \times 8\) which results to \(72\). So, \((3^{2} \cdot 2^{3})\) simplifies to \(72\).

Step 2: Calculating the power of 72.

Next, the original equation becomes \(72^{3}\). This denotes \(72 \times 72 \times 72\). Calculate this and it gives \(373248\).

Step 3: Stating the Final Result

The final result is then \(373248\), which is the evaluated and simplified calculation of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about finding ways to make expressions easier to handle or understand. In this case, we are dealing with an expression that involves exponents and multiplication, specifically \((3^{2} \cdot 2^{3})\). Simplifying this is the first step towards evaluating it.

• Start with the innermost expressions: Calculate the powers \(3^{2}\) and \(2^{3}\).
• \(3^{2}\) means multiplying three by itself, which is \(9\).
• \(2^{3}\) translates to multiplying two by itself three times, which totals \(8\).
• Next, multiply the results: \(9 \times 8 = 72\). This is the simplified result inside the parentheses.

Simplifying expressions like this makes it easier to see what we are working with before moving to more complex operations.

Order of Operations

The order of operations is a key concept in mathematics to make sure calculations are done consistently and correctly. To remember the correct order, think of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). This tells us the order to approach calculations in any expression.

• In our expression \((3^{2} \cdot 2^{3})^{3}\), the first step is to work inside the parentheses: \(3^{2} \cdot 2^{3}\).
• Evaluate powers first, making it \(9 \times 8\). When calculated, it becomes \(72\).
• Finally, apply the exponent of \(3\) to this result, \(72^{3}\).

By following the order of operations, we ensure that our final answer is correct, which in this case is \(72 \times 72 \times 72 = 373248\). It proves crucial for accuracy, especially with more complex expressions.

Evaluating Expressions

Evaluating expressions means putting all the pieces together to find the numerical value of the expression. After simplifying and using the correct order of operations, we can evaluate the expression by tackling it step-by-step.

• We previously simplified the core expression to \(72\).
• This simplified piece is then raised to the power of three in the original equation: \(72^{3}\).
• To evaluate \(72^{3}\), consider it as \(72 \times 72 \times 72\).
• Computing this, possibly with the help of a calculator, results in \(373248\).

Evaluating expressions concludes the problem-solving process, where our aim is to get an exact number value, giving a concrete outcome to the abstract expression. This approach ensures accuracy and clear understanding at every step of mathematical problems involving exponents and other operations.