Problem 6 Simplify each square root. $$ ... [FREE SOLUTION]

91影视

Elementary Algebra

Wade Ellis, Jr.

$Math Studyset 91影视 Explanations$ Math

2010 Edition

Chapter 9: Problem 6

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

Short Answer

Expert verified

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

Step by step solution

Rewrite Terms as Power of 2 Factors

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

Apply the Product of Roots Property

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

Calculate the Square Roots of the Terms with Even Exponents

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

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: $$ \sqrt{w^{7} z^{3} k^{13}} = w^{3}\sqrt{w} \cdot z \sqrt{z} \cdot k^{6}\sqrt{k} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Rewrite Terms as Power of 2 Factors

Apply the Product of Roots Property

Calculate the Square Roots of the Terms with Even Exponents

Combine the Simplified Square Roots

Key Concepts

Simplifying Radicals

Square Roots

Exponent Laws

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