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Chapter 7: Problem 87

Simplify. $$(-2 \sqrt{x y^{2} z})^{2}$$

Short Answer

Expert verified
4 x y^{2} z

Step by step solution

01

Understand the Expression

The expression given is \((-2 \sqrt{x y^{2} z})^{2}\). This involves a square of a product of terms, including a square root.
02

Apply the Square to Each Term

To simplify, square each component individually inside the parentheses. The expression \((-2 \sqrt{x y^{2} z})^{2}\) can be rewritten as \((-2)^{2} \cdot (\sqrt{x y^{2} z})^{2}\).
03

Square the Constant Term

Calculate \((-2)^{2}\). \((-2)^{2} = 4\).
04

Square the Square Root Term

Next, evaluate \((\sqrt{x y^{2} z})^{2}\). The square of a square root simply removes the square root, so \((\sqrt{x y^{2} z})^{2} = x y^{2} z\).
05

Combine and Simplify

Multiply the results from Step 3 and Step 4 together. Therefore, \((-2 \sqrt{x y^{2} z})^{2} = 4 \cdot (x y^{2} z) = 4 x y^{2} z\).

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Key Concepts

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

Simplifying Square Roots
Square roots can be tricky, but understanding them is essential for simplifying algebraic expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, \(\text{鈭�4}\) is 2 because \(2 \times 2 = 4\). In algebra, square roots work similarly with variables.\(\text{鈭歺}\) is a value which, when squared, equals \(x\). Here's a breakdown of working with square roots:
  • Recognize that \( \text{鈭歛} \) multiplied by \( \text{鈭歛} \) equals \(\text{a} \).
  • Combine like terms under the square root when possible, such as \( \text{鈭�(ab)} = \text{鈭歛} \times \text{鈭歜} \).
So, when we see \( (\text{鈭歺 y^{2} z})^{2} \) in the expression, squaring it simply removes the square root as shown in the steps!
Understanding Exponents
Exponents indicate the number of times a number, known as the base, is multiplied by itself. For example, \(2^{3} \) means \(2 \times 2 \times 2\). When simplifying exponents in algebra, the rules become particularly crucial. Here are some useful tips:
  • \

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