Calculating 4^1.5 without a Calculator: A Step-by-Step Guide

Many mathematical problems require the use of calculators for precise results. However, knowing some fundamental principles and methods can help compute expressions like 4^{1.5} without relying on advanced technology.

Understanding the Problem

The expression 4^{1.5} can be broken down into simpler components. The exponent 1.5 can be expressed as a fraction:

1.5 3/2

Thus, 4^{1.5} can be rewritten as:

4^{1.5} 4^{3/2} (4^{1/2})^3

Breaking Down the Calculation

To make the calculation more manageable, we can break down the expression into two steps:

Calculate the square root of 4: Raise the result to the power of 3:

Step 1: Calculate the Square Root of 4

The square root of 4 is:

4^{1/2} √4 2

Step 2: Raise the Result to the Power of 3

Now, we raise 2 to the power of 3:

(4^{1/2})^3 2^3 8

Therefore, the value of 4^{1.5} is:

boxed{8}

Alternative Methods for Calculation

There are multiple ways to approach this calculation without a calculator. Here are a few examples:

Example 1: 4^{1.5} 2^2^{1.5}

First, raise 2 to the power of 2, then to the power of 1.5:

2^(2×1.5) 2^3 8

Example 2: 4^{1.5} 4^3/2

Another way to express the calculation is:

4^3/2 4 × 4^{1/2} 4 × 2 8

Example 3: Using the Display Function of a Calculator

For a quick and efficient method, you can directly input the expression into a calculator's display:

Enter 4, then press the exponent button (^), then enter 1.5, and finally press the equals button ().

Exploring the Concept Further

It's worth noting that the expression 4^{1.5} can also be interpreted in terms of roots and exponents:

4^{1.5} 4^{3/2} (4^{1/2})^3

This is the same as taking the square root of 4 and then raising it to the power of 3. The result is:

8

The Double-Root and Double-Cube Nature of the Expression

It's important to understand that expressions with fractional exponents can have both positive and negative roots. Therefore, we can also express:

4^{1.5} 4^{3/2} (4^{1/2})^3

As:

(-4^{1/2})^3 -8

However, in most practical applications, the positive value is the one used.

Alternative Calculation Techniques

Using logarithms and anti-logarithms is another method to solve this expression:

Sets 4^{1/2} x

Then,

log_{10} x dfrac{log_{10} 4}{2}

10^{dfrac{log_{10} 4}{2}} x

Using an anti-logarithm to compute the 10 power decimal number, we can find x.

Practical Approximation Techniques

For a more hands-on approach, one can approximate 1.5 as the sum of simple fractions, such as 1/4, 1/32, 1/64, and 1/256. By performing straightforward square roots and multiplications, a close approximation to 8 can be obtained.

For instance:

8^{1/3} 2 and 2^2 4

Or,

-8^{1/3} -2 and -2^2 4

This shows how the process works in reverse, confirming the result.