/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Evaluate each expression. $$8+... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 47

Evaluate each expression. $$8+2 \sqrt{5^{2}-3^{2}}$$

Short Answer

Expert verified
16

Step by step solution

01

- Evaluate the exponents

First, evaluate the exponents inside the square root: \(5^2\) and \(3^2\). This gives us 25 and 9, respectively.
02

- Subtract the results

Next, subtract the smaller number from the larger number inside the square root: \(25 - 9 = 16\).
03

- Simplify the square root

Now, take the square root of 16: \(\sqrt{16} = 4\).
04

- Multiply and add

Multiply the result from the square root by 2: \(2 \times 4 = 8\). Finally, add this result to the initial 8: \(8 + 8 = 16\).

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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, typically a square root, simplifies when the number under the root is a perfect square. For instance, \(\text{鈭�(16)}\) leads to 4 because 16 is a perfect square (4 squared is 16). Simplifying radicals helps to make expressions easier to work with.
In the provided exercise, we simplified \( \text{鈭�(25 - 9)}\) to get \( \text{鈭�16} \). Since 16 is a perfect square, this simplifies directly to 4.
Order of Operations
The order of operations is a set of rules that dictates the correct sequence to evaluate a mathematical expression. Remember the acronym PEMDAS:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Following this order ensures accurate results. In our example, we first handled the operations inside the parentheses \(5^2 - 3^2\), calculated the exponentiation, and then performed subtraction before moving on to other operations.
Exponentiation
Exponentiation refers to raising a number to the power of another number, typically written as \(a^b\). This signifies that 'a' is multiplied by itself 'b' times. For example, \(5^2\) means 5 multiplied by itself, giving 25. Likewise, \(3^2\) equates to 3 multiplied by itself, resulting in 9.
In our exercise, the expressions \(5^2\) and \(3^2\) were evaluated first to obtain 25 and 9, respectively, before proceeding with further simplification.

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Most popular questions from this chapter

Simplify. $$ 3(1-x y)-2(x y-5)-(35-x y) $$

Simplify each expression. $$ x-0.03(x+500) $$

Simplify. $$ 2\left(x^{2}-3\right)-\left(6 x^{2}-2\right)+2\left(-7 x^{2}-4\right) $$

Find each product or quotient. $$ (-2 x)(-5 x) $$

Cooperative learning Work in a small group to make a list of the real numbers of the form \(\sqrt{n},\) where \(n\) is a natural number between 1 and 100 inclusive. Decide on a method for determining which of these numbers are rational and find them. Compare your group's method and results with other groups' work.
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