Problem 71 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt[4]{32}+5 \sqrt[4]{2}-\sqrt[4]{162} $$

Step-by-Step Solution

Answer

The simplified expression is \(4 \sqrt[4]{2}\).

Step 1: Express Radicals in Terms of Powers

Start by expressing each radical as a power of 2. - \( \sqrt[4]{32} = 32^{1/4} \)- \( \sqrt[4]{2} = 2^{1/4} \)- \( \sqrt[4]{162} = 162^{1/4} \)

Step 2: Break Down the Radicals

Convert each number to a form that's easier to combine by representing them as powers of their smallest base factor:- \( 32 = 2^5 \) so \( 32^{1/4} = (2^5)^{1/4} = 2^{5/4} \) - \( 2^{1/4} \) is already simplified. - \( 162 = 2 imes 81 = 2^1 imes (3^4) \), hence \( 162^{1/4} = (2 \times 3^4)^{1/4} = 2^{1/4} \times 3 \)

Step 3: Simplify the Expression

Now, convert each term to the base of 2 wherever possible:- Using Step 2, \( 32^{1/4} = 2^{5/4} = 2^{1/4} \times 2^{1} = \sqrt[4]{2} \times 2 \approx 2 \times \sqrt[4]{2} \) - \( 5 \sqrt[4]{2} \) remains as is.- \( (2^{1/4} \times 3) \) remains as is for now.

Step 4: Combine the Like Radicals

Combine terms that have the same radical base:- The terms \( 2 \sqrt[4]{2} \) and \( 5 \sqrt[4]{2} \) can be added since they are like terms: \[ 2 \sqrt[4]{2} + 5 \sqrt[4]{2} = 7 \sqrt[4]{2} \]- Subtract the remaining term from \( 7 \sqrt[4]{2} \): \[ 7 \sqrt[4]{2} - \sqrt[4]{162} = 7 \sqrt[4]{2} - 2^{1/4} \times 3 \]

Step 5: Final Simplified Expression

The simplified expression is as close as you can get unless exact numeric evaluation is done:The final simplified expression after combining like terms and making adjustments as necessary is:\[ 7 \sqrt[4]{2} - 3 \cdot \sqrt[4]{2} = 4 \sqrt[4]{2} \]

Key Concepts

Combining Like Terms

When simplifying expressions, combining like terms is a key step. Like terms have the same variables raised to the same power. In radical expressions, this means that the radicals have the same base and index. For instance, if we look at the given expression, \(2 \sqrt[4]{2}\) and \(5 \sqrt[4]{2}\) are like terms because they both contain the radical \(\sqrt[4]{2}\). You can think of this process as merging similar items in arithmetic.

• First, identify the terms that have the same radical.
• Next, add or subtract the coefficients (the numbers in front of the radicals).
• This results in a simpler expression.

Using our example, \(2 \sqrt[4]{2} + 5 \sqrt[4]{2} = 7 \sqrt[4]{2}\). This makes handling these expressions easier, reducing the risk of errors and often leading to a more insightful solution.

Radical Expressions

Radical expressions involve roots of numbers. They're often presented as square roots, cube roots, or even fourth roots, like in our problem. Understanding how to manipulate these expressions makes them less intimidating. When simplifying them, you start by expressing the numbers under the radical as a product of prime factors. This step is crucial because it lets you break down the radical into simpler forms.

• Convert numbers under the radical to their prime factor form.
• Apply the universal rule of roots: \(\sqrt[n]{a^n} = a\).
• Identify parts of the radical that can be simplified.

In the exercise, we expressed \(32^{1/4}\) and \(162^{1/4}\) in terms of factors of 2 and 3, allowing us to simplify and find like terms. By following these steps, you can unravel complex radical expressions effectively.

Exponents and Powers

Exponents and powers are integral in simplifying radical expressions. They provide a handy way to express repeated multiplication, making them very useful in algebra. With exponents, you can express roots as fractions, which is especially useful for rooting expressions.

• An exponent of \(1/n\) translates to an \(n^{th}\) root.
• Use properties of exponents, like \((a^m)^{n} = a^{m \cdot n}\), to adjust expressions as needed.
• Simplify by reducing the expression to its simplest form.

For example, \(32^{1/4}\) can be calculated by finding the fourth root, which can also be expressed as \((2^5)^{1/4} = 2^{5/4}\). In doing so, you're able to create expressions that are easier to manage and compare, especially when combining like terms. Understanding these concepts thoroughly aids in performing accurate and efficient simplifications.