Problem 25 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt[6]{64}\)

Step-by-Step Solution

Answer

The value of \(\sqrt[6]{64}\) is 2.

Step 1: Understand the Radical Expression

The expression \(\sqrt[6]{64}\) is asking you to find the sixth root of 64. This means we need to find a number which, when multiplied by itself six times, equals 64.

Step 2: Rewrite as an Exponent

Recognize that finding \(\sqrt[6]{64}\) is equivalent to raising 64 to the power of \(\frac{1}{6}\): \(64^{\frac{1}{6}}\).

Step 3: Break Down 64 into Factors

Notice that 64 is a perfect power of 2. It can be written as \(2^6\) because \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).

Step 4: Apply the Exponent Law

Using the property \((a^m)^n = a^{m \cdot n}\), rewrite \((2^6)^{\frac{1}{6}}\). This simplifies to \(2^{6 \cdot \frac{1}{6}} = 2^1 = 2\).

Step 5: Verify by Multiplying

To ensure correctness, verify by calculating \(2^6 = 64\). Since \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\), our solution is verified.

Key Concepts

Sixth Root

The term 'sixth root' refers to finding a number that, when multiplied by itself six times, gives another number. In simpler words, if you have the expression \(\sqrt[6]{64}\), you need to determine which number, when used as a factor six times, results in 64. Think of it like reverse multiplication over six steps.

• If you multiply a number by itself once, it’s squared.
• If you multiply it twice, it’s cubed.
• Here, we multiply it six times, leading to the sixth root.

For example, since \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\), the sixth root of 64 is 2. This process helps in narrowing down the sixth root by analyzing the exponents and factorization.

Exponent Laws

Exponent laws are like shortcuts in mathematics that simplify calculations when dealing with powers. Knowing these laws is essential, especially for simplifying expressions like \((2^6)^{\frac{1}{6}}\). The key law used here is \((a^m)^n = a^{m \cdot n}\).
This law states that if a number raised to an exponent is further raised to another exponent, you can multiply the exponents. In the case of finding the sixth root:\[ (2^6)^{\frac{1}{6}} = 2^{6 \cdot \frac{1}{6}} = 2^1 = 2. \]

• This simplification means you multiply 6 by \(\frac{1}{6}\), resulting in 1, as any number to the power of 1 is itself.

Exponent laws make complex calculations easier and faster, which is helpful when working in algebra and more advanced mathematics.

Perfect Powers

A perfect power is a number that can be expressed as an integer power of another integer. In other words, it can be rewritten by raising a base to a certain exponent. For instance, 64 is a perfect power because it is \(2^6\).
Why is this important? Identifying perfect powers helps simplify roots and exponents more easily. They also ensure that mathematical transformations are much more direct. Here's how it applies to our example:

• Recognizing 64 as \(2^6\) allows us to transition smoothly through exponent laws.
• This recognition is the foundation for simplifying roots, such as transitioning from exponential to root expressions.

So, once you know how to identify a perfect power, managing root and exponential problems becomes less daunting.

Simplifying Radicals

Simplifying radicals is a process that involves expressing a root in its simplest form. Radicals can often complicate expressions, but using steps like factorization and exponent laws can make simplification straightforward and efficient.
For the expression \(\sqrt[6]{64}\), simplifying radicals involves:

• Recognizing the inside number (64) as a perfect power \(2^6\).
• Applying the exponent law, which lets us simplify \((2^6)^{\frac{1}{6}}\) to \(2\).

When terms inside the radical can be expressed as powers of their simplest form, simplifying becomes easier. As radicals become more complex, these techniques offer essential aid to peel back layers of the expression until they are simple and manageable.