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Chapter 10: Problem 43

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$(\sqrt{25-9})^{2}$$

Short Answer

Expert verified
16

Step by step solution

01

- Simplify Inside the Square Root

Evaluate the expression inside the square root, \( \sqrt{25-9} \). Subtract 9 from 25 to get 16.
02

- Evaluate the Square Root

Find the square root of 16, which is 4, because \( 4 \times 4 = 16 \). So \( \sqrt{16} = 4 \).
03

- Square the Result

Square the result obtained in the previous step. Therefore, \( 4 ^{2} = 16 \).

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Key Concepts

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

Square Roots
Square roots help us find which number, when multiplied by itself, gives us the original number. In our example, we're working with \(\sqrt{25-9}\). The first step is to simplify inside the square root. Here, we subtract 9 from 25 to get 16, making the expression \(\sqrt{16}\). Remember, the square root of 16 is the number that, when multiplied by itself, equals 16. This number is 4, because \(4 \cdot 4 = 16\). Therefore, \(\sqrt{16} = 4\).
Basic Arithmetic
Basic arithmetic includes addition, subtraction, multiplication, and division. To understand our exercise, we first need to do subtraction inside the square root. We started with \(\sqrt{25-9}\). Subtracting 9 from 25 involves basic arithmetic: \(25-9=16\). This helps simplify our problem to \(\sqrt{16}\). Knowing how to perform these operations quickly and accurately is crucial, as they lay the foundation for more complex algebraic manipulations.
Order of Operations
The order of operations is a set of rules that dictate the sequence in which calculations should be performed. In our problem, \( (\sqrt{25-9})^2\), we follow these steps:
  • First, simplify inside the square root: \(\sqrt{25-9} = \sqrt{16}\).
  • Next, evaluate the square root: \(\sqrt{16} = 4\).
  • Finally, perform the squaring operation: \( 4^2 = 16\).
Following this order ensures that we accurately evaluate and simplify expressions, avoiding mistakes and confusion.

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Most popular questions from this chapter

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{3}(\sqrt{5}+\sqrt{6})$$

Use a calculator to give decimal approximations rounded to one, two, and three places. $$\sqrt{260}$$

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{115}$$

Use a calculator to give decimal approximations rounded to one, two, and three places. $$\sqrt{23}$$

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{872}$$
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