Problem 10 - Fundamentals of Mathematics

Question

Find the following roots using only a knowledge of multiplication. \(\sqrt[6]{64}\)

Step-by-Step Solution

Answer

The 6th root of 64 is 2.

Step 1: Understand the Problem

We need to find the 6th root of 64. This means we are looking for a number which, when multiplied by itself six times, equals 64. Mathematically, this is represented by \(x^6 = 64\).

Step 2: Factorize 64

To simplify finding the 6th root, start by factorizing 64. Notice that 64 is a power of 2: \(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6\). This means that 64 is 2 raised to the power of 6.

Step 3: Identify the 6th Root

Since 64 is \(2^6\), the 6th root of 64 is the base of this expression, which is 2, because \(2^6 = 64\).

Step 4: Verify the Solution

To ensure the solution is correct, verify by calculating \(2^6\): \(2 \times 2 = 4\)\(4 \times 2 = 8\)\(8 \times 2 = 16\)\(16 \times 2 = 32\)\(32 \times 2 = 64\). The multiplication confirms \(2^6 = 64\), which proves that our solution is correct.

Key Concepts

Factorization

Factorization is the process of breaking down a number into its factors, which are smaller numbers that multiply together to make the original. When dealing with nth roots, factorization helps simplify the process of finding the root. In our exercise, we factorized 64. We started by recognizing that 64 is a product of the same number.

• 64 = 2 × 2 × 2 × 2 × 2 × 2
• This results in 64 = 2⁶

Breaking down numbers into prime factors can make calculations more manageable when working with roots. Here, because 64 is expressed as a power of 2, it directly shows us that the 6th root involves the base number in the prime factorization. Factorization simplifies challenging multiplication into straightforward multiplication tables. Becoming comfortable with factorization supports both quick mental math and understanding complex mathematical operations.

Powers and Exponents

Powers and exponents represent repeated multiplication of the same number. An exponent (or power) is placed on top right of a base number, showing how many times this base is multiplied by itself. In the expression 2⁶, 2 is the base, and the exponent is 6.

• Powers: Base number with an exponent, e.g., 2⁶
• Exponent: Number that shows repeated multiplication, e.g., 6

When considering nth roots, like the 6th root of 64, the idea is to "reverse" this process of repeated multiplication. The root tells us that we need to find a number which, when raised to the 6th power, results in 64. Seeing that 64 is already written as 2⁶ simplifies identifying the root: it's the number in the position of the base. Exponentiation helps express large products in a concise and efficient manner, crucial for handling complex calculations.

Multiplication Verification

Multiplication verification is a key step to ensure that our understanding and solution are correct. Once we solve for a root, verifying by multiplying the number by itself repeatedly checks the accuracy of our solution. In this exercise, we confirmed that 2 is indeed the 6th root of 64 by carrying out the multiplication:

• Start: 2 × 2 = 4
• Then: 4 × 2 = 8
• Next: 8 × 2 = 16
• So on: 16 × 2 = 32
• Finally: 32 × 2 = 64

This sequence accurately reconstructs the original number by verifying that each step adheres to the process of multiplying the same base, showcasing correct computation. By doing this, any errors in earlier steps can be caught and rectified, ensuring the integrity of your mathematical solution.