Problem 458 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[4]{16 x^{8}}\) (b) \(\sqrt[6]{64 y^{12}}\)

Step-by-Step Solution

Answer

a) 2x² b) 2y²

Step 1: Simplify the Expression (a) \( \sqrt[4]{16 x^{8}} \)

First, factor 16 as a power of 4: \(16 = 2^4\). Therefore, \( \sqrt[4]{16} = \sqrt[4]{2^4} = 2\). For the variable part, use the property of radicals: \( \sqrt[4]{x^8} = (x^8)^{1/4} = x^{8/4} = x^2 \). Thus, the simplified form is \(2 x^2 \).

Step 2: Simplify the Expression (b) \( \sqrt[6]{64 y^{12}} \)

First, factor 64 as a power of 6: \(64 = 2^6\). Therefore, \( \sqrt[6]{64} = \sqrt[6]{2^6} = 2\). For the variable part, use the property of radicals: \( \sqrt[6]{y^{12}} = (y^{12})^{1/6} = y^{12/6} = y^2 \). Thus, the simplified form is \(2 y^2 \).

Key Concepts

Understanding Radicals

Radicals are the opposite of exponents. When you see a radical, it asks for which number, when raised to a certain power, gives the number inside the radical. For example, \(\sqrt{9}\) asks which number multiplied by itself gives 9, and the answer is 3. Radicals can also involve higher roots, like cube roots (\(\sqrt[3]{...}\)) and fourth roots (\(\sqrt[4]{...}\)).

In the exercise, simplifying \(\sqrt[4]{16 x^{8}}\) and \(\sqrt[6]{64 y^{12}}\) involves higher roots. Since radicals and exponents share properties, we can use the relationship between them for simplification.

Working with Exponents

Exponents tell you how many times a number should be multiplied by itself. For example, \(3^2 = 3 \times 3 = 9\). There are also fractional exponents, where the denominator signifies the root. So, \(x^{1/n}\) is the same as \(\sqrt[n]{x}\).

In our exercise, it's crucial to convert the radicals into their exponent form. For \(\sqrt[4]{16 x^{8}}\), we express 16 as \(2^4\) and the fourth root becomes \(2^{4/4} = 2\). For the variable part, \(x^8\) under the fourth root becomes \(x^{8/4} = x^2\). The same logic applies to \(64 y^{12}\), where converting 64 to \(2^6\) and taking the sixth root gives \(2\). For the variable part, \(y^{12}\) under the sixth root simplifies to \(y^2\).

Algebraic Simplification

Algebraic simplification is about making expressions as simple as possible. Simplifying radicals often involves expressing numbers as powers and reducing them using exponent rules.

Let's break down the steps:

• Identify the base number and its exponent
• Express the base number as a power if not already done
• Apply the root by converting it to a fractional exponent
• Simplify the exponents.

In the first example, \(\sqrt[4]{16} \equiv 2\) and \(\sqrt[4]{x^8} \equiv x^2\), so combining them gives \(2x^2\).
In the second example, \(\sqrt[6]{64} \equiv 2\) and \(\sqrt[6]{y^{12}} \equiv y^2\). Combining them results in \(2y^2\). Thus, simplifying radicals involves careful application of exponent rules, making algebraic expressions easier to handle.