Problem 5 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ -2(27)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is -18.

Step 1: Understand the Expression

We're given the expression \(-2(27)^{2/3}\). This means we need to first evaluate \(27^{2/3}\) and then multiply the result by \-2\.

Step 2: Evaluate the Fractional Exponent

The expression \(27^{2/3}\) involves a fractional exponent. The base is 27, which can be rewritten as \(3^3\) because \(3 \times 3 \times 3 = 27\). The fractional exponent \({2/3}\) means you raise the expression to the power of 2 and then take the cube root, or vice versa.

Step 3: Simplify the Expression using Cube Root

First, calculate the cube root of 27, which is \((27^{1/3} = 3)\). This is because \(3^3 = 27\).

Step 4: Raise the Result to the Power 2

Now, square the result from the previous step: \(3^2 = 9\). This means \(27^{2/3} = 9\).

Step 5: Apply the Negative Sign and Multiplication

The original expression is \(-2 imes (27)^{2/3}\). Substitute the simplified value \(9\) for \(27^{2/3}\), resulting in \(-2 imes 9 = -18\).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express powers and roots together. They simplify expressions that involve both operations. For example, the expression \( x^{2/3} \) can be broken down into two steps: raising \( x \) to the power of 2 and then finding the cube root of the result, or performing these steps in the reverse order.

• The numerator of the fractional exponent (here, it is 2) indicates the power to which the number is raised.
• The denominator (in this case, 3) indicates the root to be taken.

In our problem, the expression \( 27^{2/3} \) tells us to take the cube root of 27 first and then square the result. This method simplifies problems without having to rely on calculators, making math more approachable.

Cube Roots

Cube roots are the reverse operation of cubing a number. They answer the question: "What number, multiplied by itself three times, gives me this original number?" Calculating cube roots can reduce numbers to simpler forms, keeping expressions neat.For 27, we know that \( 3^3 = 27 \), so the cube root of 27 is 3, expressed as \( 27^{1/3} = 3 \).When working with cube roots, it's helpful to:

• Understand the cube of basic numbers (like 1, 2, 3, etc.) for quick mental math.
• Use the cube root operation to simplify fractional exponents as shown in this problem.

This technique is invaluable when dealing with real-world problems involving volume and capacity, as they often rely on cubed measurements.

Negative Numbers

Negative numbers are numbers with a value less than zero. They are crucial for expressing debts, temperatures below freezing, or elevations below sea level. When dealing with expressions, they represent a direction or distance on a number line.For the given exercise, there is a multiplication with a negative number right before the variable. This means:

• Once you simplify \( 27^{2/3} \) to 9, you apply the negative multiplier \(-2\).
• The final result is \(-2 \times 9 = -18\).

Dealing with negative numbers involves understanding that multiplying by a negative reverses the direction on the number line, which is why the result is negative. It introduces an element of subtraction or reversal in practical scenarios, such as transitioning below a reference point.