Problem 63 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

311.967

Step 1: Understand the Expression

The expression we need to evaluate is \(46^{1.5}\). This expression represents 46 raised to the power of 1.5.

Step 2: Interpret the Power

The power 1.5 can be rewritten as \(\frac{3}{2}\), which means that 46 is raised to the power of 3, and then the square root is taken, or equivalently, it can also be computed as the square root of 46 raised to the power of 3.

Step 3: Calculate the Cube

First, let's calculate 46 raised to the power of 3. Essentially, this is done by multiplying 46 by itself three times: \[46^3 = 46 \times 46 \times 46 = 97336.\]

Step 4: Find the Square Root

Next, find the square root of the result from Step 3. So, we calculate:\[\sqrt{97336} \approx 311.967.\]

Step 5: Round the Result

Finally, round the calculated square root to the nearest thousandth. The result is already given to three decimal places:\[311.967.\]

Key Concepts

Understanding Exponentiation

Exponentiation is the mathematical process of raising a number, known as the base, to a power (or exponent). In the expression \( 46^{1.5} \), 46 is our base, and 1.5 is the exponent. When you see an exponent, think of it as a repeated multiplication. Essentially, you're multiplying the base by itself a number of times indicated by the exponent.
However, when the exponent is a non-whole number, like 1.5, it means we must also apply root calculations. It’s helpful to rewrite 1.5 as \( \frac{3}{2} \). This shows that you're dealing not only with powers but also with roots.
Here's a key point about exponents:

• An integer exponent (like 3) means repeat multiplication.
• A fractional exponent (like \(\frac{1}{2}\)) indicates a root operation.
• A mixed number exponent (like 1.5 or \(\frac{3}{2}\)) combines both these operations.

This is the foundational concept behind the expression \(46^{1.5}\) and involves both raising to a power and taking a root.

Root Calculation

Root calculation involves finding a number that, when raised to a particular power, gives you the original number. In terms of our exercise on \(46^{1.5}\), we must find a balance between powering and rooting. This is because 1.5 or \(\frac{3}{2}\) represents a mixed operation: raising to the cubic power followed by a square root.
Typically:

• To find \(x^{\frac{n}{m}}\), you can directly calculate \(x^n\) and then find the \(m\)-th root.

For \(46^{\frac{3}{2}}\):

• First, calculate \(46^3=97336\). This is an example of exponentiation.
• Then, find the square root: \(\sqrt{97336} = 311.967\).

Understanding both steps in this hybrid exponent/root expression is critical when approximating decimal values.

Rounding Numbers

Rounding numbers is the method of adjusting numbers to make them easier to work with, particularly when precision beyond a certain point is unnecessary. In our exercise, we're rounding to the nearest thousandth, or three decimal places. This is how you'd approach it:

• Look at the fourth decimal place to determine if you round up or down.
• If this digit is 5 or more, round up the third decimal place by one.
• If it's less than 5, keep the third decimal place as is.

Applying this to \(\sqrt{97336} \approx 311.967\), since there's no decimal after the third place in this instance, the number is already suited to rounding off.
Rounding helps us simplify numbers for easier comprehension and usability, especially in fields requiring quick estimations and concise data presentation.