Problem 43 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[6]{64 x^{12}} $$

Step-by-Step Solution

Answer

The root is \(2x^2\).

Step 1: Understand the Problem

We need to find the sixth root of the expression \( \sqrt[6]{64 x^{12}} \). This involves simplifying the expression under the root.

Step 2: Break Down the Expression

We recognize \( 64 \) as a perfect square and perfect sixth power: \( 64 = 2^6 \). Thus, \( 64 = (2^6) \). The expression now is \( \sqrt[6]{2^6 \, x^{12}} \).

Step 3: Apply the Root

To apply the sixth root, use the property \( \sqrt[6]{a^6} = a \). This can be applied separately to each part: \( \sqrt[6]{2^6} = 2 \) and \( \sqrt[6]{x^{12}} = x^{12/6} = x^2 \).

Step 4: Simplify the Result

Combine the simplified parts: \( 2 \times x^2 = 2x^2 \).

Step 5: Verify the Solution

Double check by plugging back: \((2x^2)^6 = 2^6 x^{12} = 64x^{12},\) which matches the original expression under the root.

Key Concepts

Simplifying Expressions

Simplifying expressions is like decluttering a room or organizing a messy desk. The goal is to make the mathematical expression as simple and clear as possible. This often involves reducing terms and finding common factors.
When simplifying expressions that involve roots, such as the one in the exercise, it's important to identify whether numbers and variables can be expressed in a more concise form.

• Look for numbers that are perfect powers (like 64 in our exercise, which is a perfect sixth power).
• Break down variables with exponents to see if they can be simplified (like the exponent 12 for x).

Through this process, we can express the components of the expression in a way that makes it easier to solve or further manipulate.

Nth Roots

The concept of nth roots can initially seem complex, but it's essentially about finding a number that, when raised to the power of n, gives back the original number.
For example, the 6th root (denoted as \(\sqrt[6]{...}\)) asks us to find a number that, when multiplied by itself six times, equals the original quantity.

• If we apply the nth root to numbers like 64, we find that \(2^6 = 64\), making 2 the 6th root of 64.
• For variables, when the exponent can be divided by n, the process is straightforward. This is why \(\sqrt[6]{x^{12}} = x^{12/6} = x^2\).

Understanding nth roots is crucial for working with indices and simplifying under-root expressions efficiently.

Exponents

Exponents are the language of repeated multiplication. They tell us how many times a number or variable (the base) is multiplied by itself.
In algebra, managing exponents is key to simplifying expressions and solving equations. Recognizing patterns, like perfect squares or other powers, can help in solving problems quickly.

• In the exercise, we identified that 64 can be written as \( 2^6 \), which made it easier to apply the sixth root.
• The exponent rules also helped break down \( x^{12} \) into smaller, more manageable parts for simplification.

By mastering exponents, we can transform complex algebraic expressions into simpler forms.

Perfect Powers

Perfect powers are numbers or variables raised to an integer exponent. Recognizing them is significant for simplifying expressions because they allow us to use root properties efficiently.
64 is a perfect power because it can be expressed as \(2^6\). This is not just a neat trick; it's important for applying roots properly.

• Numbers like 64, which fit this pattern, are easier to work with because they have specific and predictable roots.
• Understanding variables as perfect powers helps in reducing algebraic expressions, such as turning \( x^{12} \) into \( (x^2)^6 \).

Spotting perfect powers lets us take shortcuts in arithmetic and algebra, making math much less intimidating.