Problem 63 - A Graphical Approach to College Algebra

Question

Find a decimal approximation of each root or power. Round answers to the nearest thousandth. $$46^{1.5}$$

Step-by-Step Solution

Answer

46^{1.5} \approx 311.867.

Step 1: Understand the Power

The expression \(46^{1.5}\) indicates raising the number 46 to the power of 1.5. This is the same as \(46\) raised to the power of \(\frac{3}{2}\).

Step 2: Break Down the Expression

The exponent \(\frac{3}{2}\) can be expressed as \(46^{\frac{3}{2}} = (46^3)^{\frac{1}{2}}\). This means we will first calculate \(46^3\) and then take the square root of that result.

Step 3: Calculate \(46^3\)

First, compute \(46^3\). This is \(46 \times 46 \times 46 = 97336\).

Step 4: Calculate the Square Root

Now, find the square root of 97336, which is written mathematically as \(\sqrt{97336}\). Compute this to get \(311.867\) when rounded to the nearest thousandth.

Step 5: Round to the Nearest Thousandth

The final result of \(46^{1.5}\) after performing the computations is approximately \(311.867\) when rounded to the nearest thousandth.

Key Concepts

Roots and Powers

When working with roots and powers, you often encounter expressions involving fractional exponents. These expressions might look complex, but they're manageable by breaking them into more familiar steps. Fractional exponents, like the \(1.5\) in \(46^{1.5}\), essentially combine roots and powers in one operation. Here, it's equivalent to \(46^{\frac{3}{2}}\), or \[\sqrt[2]{46^3}\].

Let's visualize this step-by-step:

• The numerator of the fractional exponent (3) serves as the power.
• The denominator (2) is the root you need to find after calculating the power.

This approach emphasizes how powers and roots intertwine, simplifying a once intimidating problem into manageable parts.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. In the expression \(46^{1.5}\), the number 46 is the base, and 1.5 is the exponent.

Here's how to think about exponents:

• An exponent of 2, as in \(46^2\), means \(46 \times 46\).
• An exponent of 3/2, as seen in \(46^{3/2}\), involves a combination of multiplying 46 three times and then finding the square root.

Exponents allow us to write these repeated multiplications succinctly. They simplify complex calculations, which is especially useful in algebra and higher mathematics. By understanding how to manipulate exponents, you gain a powerful tool for solving mathematical problems.

Square Root Calculation

Square root calculation comes into play when dealing with fractional exponents. In the case of \(46^{1.5}\), after computing \(46^3\), we found ourselves needing to determine the square root of 97336.

Here’s a quick breakdown:

• The square root operation is represented by \(\sqrt{}\).
• In \(\sqrt{97336}\), you're finding what number, when multiplied by itself, equals 97336.

Performing the square root gives us approximately 311.867. Leveraging a calculator simplifies this step, ensuring precision in results. Understanding how to calculate the square root is essential for tackling problems involving roots and powers, making these tasks far less daunting.