Problem 513 - Elementary Algebra

Question

In the following exercises, simplify. \(\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}\)

Step-by-Step Solution

Answer

The simplified form is \[ 2 x \times \root{5}{2 x}. \]

Step 1 - Simplify the Radicals

Recognize that both the numerator and the denominator are inside the fifth root, so rewrite them separately: \[ \frac{\root{5}{128 x^8}}{\root{5}{2 x^2}}. \]

Step 2 - Combine the Radicals

Combine the radicals into a single radical: \[ \root{5}{\frac{128 x^8}{2 x^2}}. \]

Step 3 - Simplify the Expression Inside the Root

Simplify the fraction inside the root: \[ \frac{128 x^8}{2 x^2} = 64 x^6. \]

Step 4 - Apply the Fifth Root

Take the fifth root of the simplified expression: \[ \root{5}{64 x^6}. \]

Step 5 - Simplify Further

Recognize that 64 is a power of 2, specifically, \[ 64 = 2^6. \] Therefore, the expression becomes: \[ \root{5}{2^6 x^6}. \]

Step 6 - Evaluate the Root

Separate the roots: \[ \root{5}{2^6} \times \root{5}{x^6}. \]

Step 7 – Simplify Each Term

Simplify each term under the radical: \[ \root{5}{2^6} = 2^{\frac{6}{5}} = 2^{1 + \frac{1}{5}} = 2 \times \root{5}{2}, \] and \[ \root{5}{x^6} = x^{\frac{6}{5}} = x^{1 + \frac{1}{5}} = x \times x^{\frac{1}{5}}. \]

Step 8 - Combine Simplified Terms

Combine the simplified terms to get the final expression: \[ 2 \times \root{5}{2} \times x \times x^{\frac{1}{5}}. \]

Key Concepts

Fifth Roots

The fifth root of a number is the value that, when multiplied by itself five times, gives the original number. In mathematics, this can be expressed using the radical symbol with a small '5' indicating the root, such as \(\root{5}{a}\). For example, the fifth root of 32 is 2 because \[2^5 = 32\].
When simplifying expressions involving fifth roots, it’s helpful to recognize if the number under the root is a power of a smaller number.
This makes it easier to simplify, as shown in the example where \(64 = 2^6\).

Algebraic Fractions

Algebraic fractions are fractions with polynomial expressions in the numerator and/or the denominator. To simplify, you must factor both the numerator and the denominator and then cancel out any common factors.

• In our problem, we start with \(\frac{128 x^8}{2 x^2}\).
• We simplify by canceling common factors: \(\frac{128}{2}\) and \(\frac{x^8}{x^2}\).
• This results in the simpler fraction \(\frac{128}{2} \times \frac{x^8}{x^2} = 64x^6\).

Remember, reducing algebraic fractions is crucial for simplifying more complex expressions.

Exponent Rules

Exponent rules help us simplify expressions where variables have powers. Key rules include the product rule \((a^m \times a^n = a^{m+n})\), the quotient rule \((\frac{a^m}{a^n} = a^{m-n})\), and the power rule \((a^{m^n} = a^{m \times n})\).
In our example, when we have \(x^8\) divided by \(x^2\), we use the quotient rule: \(x^8 \div x^2 = x^{8-2} = x^6\). From there, simplifying inside the fifth root, we use the property of powers to split the expression for easier simplification.

Simplifying Expressions

Finally, simplifying expressions involves combining like terms and reducing radicals to their simplest form. In the given solution:

• We start by combining radicals: \( \root{5} \frac{128 x^8}{2 x^2}\).
• Next, we simplify inside the root: \(\frac{128 x^8}{2 x^2} = 64x^6\).
• Then take the fifth root: \( \root{5}{64x^6} \) knowing that \(64 = 2^6\).
• Finally, separate and simplify each term: \( \root{5}{2^6} = 2^{6/5} = 2 \times \root{5}{2} and \root{5}{x^6} = x^{6/5} = x \times x^{1/5} \).

This results in the simplified expression: \2 \times \root{5}{2} \times x \times x^{1/5}.\ Remember, simplifying expressions step-by-step helps to avoid mistakes and ensures you find the correct solution.