Problem 6 - Elementary Algebra

Question

Simplify each square root. $$ \sqrt{w^{7} z^{3} k^{13}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(w^{3}\sqrt{w} \cdot z \sqrt{z} \cdot k^{6}\sqrt{k}\).

Step 1: Rewrite Terms as Power of 2 Factors

We begin by breaking down each term as power of 2 factors: $$ \sqrt{w^{7} z^{3} k^{13}} = \sqrt{w^{6} w z^{2} z k^{12} k} $$

Step 2: Apply the Product of Roots Property

Using the property, \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we get: $$ \sqrt{w^{6} w z^{2} z k^{12} k} = \sqrt{w^{6}} \cdot \sqrt{w} \cdot \sqrt{z^{2}} \cdot \sqrt{z} \cdot \sqrt{k^{12}} \cdot \sqrt{k} $$

Step 3: Calculate the Square Roots of the Terms with Even Exponents

We can now compute the square roots of the terms with even exponents: $$ \sqrt{w^{6}} = w^{3}, \quad \sqrt{z^{2}} = z, \quad \sqrt{k^{12}} = k^{6} $$

Step 4: Combine the Simplified Square Roots

Finally, we have: $$ w^{3} \cdot \sqrt{w} \cdot z \cdot \sqrt{z} \cdot k^{6} \cdot \sqrt{k} $$ The fully simplified expression is: $$ \sqrt{w^{7} z^{3} k^{13}} = w^{3}\sqrt{w} \cdot z \sqrt{z} \cdot k^{6}\sqrt{k} $$

Key Concepts

Simplifying Radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. A radical expression includes a root, such as a square root, cube root, etc. In this exercise, we focus on the square root.

• To simplify, identify whole number factors and perfect powers inside the radical.
• Next, rewrite these using exponent rules, focusing on even powers to easily remove them from under the square root.
• The product property of square roots, \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \), is particularly useful here.

Breaking down each term into factors with even exponents lets us simplify more effectively. This might mean expressing parts of the original term, like \( w^7 \), into something like \( w^6 \cdot w \), where \( w^6 \) can be further simplified outside the radical.

Square Roots

Square roots are essential in algebra and other areas of math. The square root of a number \( x \), represented as \( \sqrt{x} \), is a value that, when multiplied by itself, gives \( x \).

• Square roots help indicate the principal root of any given number or expression.
• For an expression like \( \sqrt{w^6} \), you're looking for the value that squared equals \( w^6 \).
• If you have even exponents (e.g., \( w^6 \)), they can be split in half, resulting in \( w^{6/2} \) or \( w^3 \).

In simplifying the original expression here, recognizing that \( (w^3)^2 = w^6 \) is key. This insight helps you take terms outside the radical with confidence.

Exponent Laws

Exponent laws are fundamental rules that govern operations involving powers of numbers or variables.

• The Power of a Product Rule states that \( (ab)^n = a^n \cdot b^n \).
• The Power of a Power Rule states that \( (a^m)^n = a^{m \cdot n} \); this helps in simplifying expressions significantly.
• The Multiplication of Powers Rule says \( a^m \cdot a^n = a^{m+n} \), which simplifies combined powers of the same base.
• Even and odd powers change how we simplify expressions; terms with even exponents can easily step out of a square root.

Applying these rules effectively allows terms with even exponents, such as \( w^6 \) and \( k^{12} \), to be processed into simpler forms like \( w^3 \) and \( k^6 \). This ensures the expression simplifies cleanly outside the square root.