Problem 61 - College Algebra

Question

Simplify each expression. $$\sqrt[4]{\frac{162 x^{6}}{16 x^{4}}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{3 \sqrt{2x}}{2}\).

Step 1: Simplify the Fraction Inside the Radical

First, look at the expression inside the fourth root: \(\frac{162 x^6}{16 x^4}\). Simplify this fraction by dividing both the numerator and the denominator by the greatest common factor of the coefficients and apply the laws of exponents to simplify the powers of x. The GCF of the numbers 162 and 16 is 2.

Step 2: Divide the Coefficients

Divide 162 by 2 to get 81, and divide 16 by 2 to get 8. This results in the simplified fraction \(\frac{81 x^6}{8 x^4}\).

Step 3: Simplify the Exponent of x

Apply the rule \(x^a / x^b = x^{a-b}\) to the exponents. Subtract the exponent in the denominator from the exponent in the numerator: \(x^6 / x^4 = x^{6-4} = x^2\). The fraction becomes \(\frac{81 x^2}{8}\).

Step 4: Simplify the Fourth Root

Rewrite the expression as \(\sqrt[4]{\frac{81 x^2}{8}} = \frac{\sqrt[4]{81 x^2}}{\sqrt[4]{8}}\). Find the fourth root of each part. The fourth root of 81 is 3 (since \(3^4 = 81\)), and the fourth root of \(x^2\) is \(x^{1/2}\) or \(\sqrt{x}\).

Step 5: Calculate the Fourth Root of Denominator

The fourth root of 8 is the number that, when raised to the power of 4, equals 8. Approximating, this is \(8^{1/4}\), which simplifies approximately to \(\sqrt{2}\) when expressing as a radical.

Step 6: Combine the Results

Combine the roots: \(\frac{3 \sqrt{x}}{\sqrt{2}}\). To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{2}\) to eliminate the square root from the denominator.

Step 7: Rationalize the Denominator

This multiplication results in \(\frac{3 \sqrt{x} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{3 \sqrt{2x}}{2}\). The denominator simplifies as \(\sqrt{2} \cdot \sqrt{2} = 2\).

Key Concepts

Fourth Root

The fourth root of a number is the value that satisfies the equation when raised to the fourth power. In simpler terms, if you say the fourth root of 81, you are looking for a number that multiplied by itself four times gives 81 which in this case is 3 because:\[3^4 = 81\]When you are dealing with variables, like in \( \sqrt[4]{x^2} \), the process includes turning the expression into its fractional exponent form: \( (x^2)^{1/4} = x^{2/4} = x^{1/2} \), simplifying to \( \sqrt{x} \).
The fourth root is useful in simplifying complex expressions, helping make calculations more manageable.

Rationalizing the Denominator

In algebra, especially when dealing with fractions involving radicals or roots, it is common to 'rationalize' the denominator. This means converting the denominator into a rational number (without radicals).
For example, consider the expression \( \frac{3 \sqrt{x}}{\sqrt{2}} \). The goal is to eliminate the radical \( \sqrt{2} \) from the denominator. To do this, multiply both the numerator and denominator by \( \sqrt{2} \), which effectively turns the denominator into a simpler expression:

• Multiply \( \sqrt{2} \cdot \sqrt{2} \) to get 2 (since \( \sqrt{2}^2 = 2 \)).
• The numerator becomes \( 3 \sqrt{x} \cdot \sqrt{2} = 3 \sqrt{2x} \).

After rationalizing, you obtain \( \frac{3 \sqrt{2x}}{2} \), simplifying the expression significantly.

Exponents

Exponents are a shorthand way to show how many times a number is to be multiplied by itself. For example, in the expression \( x^6 \), "6" is the exponent indicating that x is multiplied by itself six times.
When simplifying expressions containing exponents, you often apply laws of exponents:

• \( x^a \cdot x^b = x^{a+b} \)
• \( \frac{x^a}{x^b} = x^{a-b} \)
• \( (x^a)^b = x^{a \cdot b} \)

In our problem, we simplify \( x^6 \) divided by \( x^4 \) by applying \( \frac{x^6}{x^4} = x^{6-4} = x^2 \), allowing easy reduction of x terms.

Fractions

Fractions represent parts of a whole and consist of a numerator and a denominator. Simplifying fractions can sometimes involve reducing both parts by a common factor or applying other operations like division or cancellation.
In the given exercise, simplifying the fraction \( \frac{162 x^6}{16 x^4} \) involves:

• Finding the greatest common factor (GCF) of the numbers, which is 2 in this case.
• Dividing both 162 and 16 by 2, resulting in \( \frac{81 x^6}{8 x^4} \).
• Further simplifying the fraction terms by reducing the exponents of x using exponent rules.

Learning to manipulate and simplify fractions, especially involving exponents and radicals, provides key strategies in solving various algebraic expressions efficiently.