Problem 501 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[6]{64}\)

Step-by-Step Solution

Answer

2

Step 1: Identify the base number

First, recognize that 64 is a base number. It can be rewritten as a power of another number.

Step 2: Express the base number as a power

Rewrite 64 as a power of 2. So, 64 = 2^6.

Step 3: Simplify the radical expression

Now, use the property of radicals \( \sqrt[n]{a^n} = a \) . Applying this property, the 6th root of \(2^6\) is simply 2.

Key Concepts

Understanding Radicals

Radicals are a way to represent roots of numbers, similar to how exponents represent repeated multiplication. For example, the square root of a number is a radical denoted by the symbol \( \sqrt{} \). More generally, the nth root of a number is represented by \( \sqrt[n]{} \).
When you see an expression like \( \sqrt[6]{64} \), it means you are looking for the number which, when raised to the power of 6, will give you 64. In simpler terms, what number multiplied by itself 6 times equals 64?

• When dealing with radicals, remember that they are closely related to exponents.
• The notation can sometimes be tricky, but practicing makes it easier.

Generally, remembering the relation between radicals and exponents simplifies many mathematical problems.

Exploring Exponents

Exponents are a shorthand notation for expressing repeated multiplication of a number by itself. For example, \( 2^6 \) means multiplying 2 by itself 6 times: 2 × 2 × 2 × 2 × 2 × 2 = 64.
Recognizing how a number can be expressed as an exponent is crucial when working with radicals. This relationship is beneficial when simplifying radical expressions because it allows us to rewrite the problem in a form that's easier to handle.
Here are a few key points about exponents:

• An exponent like \( a^n \) means that the base 'a' is multiplied by itself 'n' times.
• Exponents and powers are interchangeable terms.
• Understanding exponents allows you to see patterns in numbers more easily, especially with radicals.

By translating a number into its exponential form, you can use properties of radicals to simplify expressions.

Demystifying Roots

Roots are the inverse operation of exponents. If you know that \( 2^6 = 64 \), then the 6th root of 64 will be 2. The general formula for this is \( \sqrt[n]{a^n} = a \).
When simplifying a root, the goal is often to rewrite it in exponential form first: it makes the operation straightforward. So, for \( \sqrt[6]{64} \), recognizing 64 as \( 2^6 \) allows you to use the property \( \sqrt[n]{a^n} \).
A few bullet points to keep in mind:

• Roots can be seen as the opposite of raising a number to a power.
• They help find the original number used in repeated multiplication to reach a given value.

By understanding how roots and exponents interact, simplifying complex expressions becomes more manageable and less intimidating.