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Chapter 10: Problem 11

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$\sqrt{64}$$

Short Answer

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Step by step solution

01

Understand the Square Root

The problem asks to evaluate the square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number.
02

Identify Perfect Squares

Identify that 64 is a perfect square. A perfect square is an integer that can be expressed as the square of another integer.
03

Find the Square Root

Determine which integer squared equals 64. We know that \(8 \times 8 = 64\). Hence, \sqrt{64} = 8\.

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Key Concepts

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

square root
A square root is a special mathematical operation. When you take the square root of a number, you're looking for a value that, when multiplied by itself, equals the original number. For example:

  • The square root of 16 is 4 because 4 脳 4 = 16.
  • The square root of 25 is 5 because 5 脳 5 = 25.
In mathematical notation, the square root of a number is written like this: \( \sqrt{ number } \) For instance, \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \). Understanding square roots is essential for various mathematical concepts, as they often appear in algebra, geometry, and calculus.

When asked to find the square root of a number, it's helpful to be familiar with perfect squares, which we'll discuss next.
perfect squares
Perfect squares are numbers that can be expressed as the product of an integer with itself. Simply put, if you can find an integer 'n' such that \( \ n \times n \ = \ k\), then 'k' is a perfect square. Here are some common examples:

  • 1 is a perfect square because \(1 \times 1 = 1 \).
  • 4 is a perfect square because \(2 \times 2 = 4 \).
  • 9 is a perfect square because \(3 \times 3 = 9 \).
  • 16 is a perfect square because \(4 \times 4 = 16 \).
Knowing these perfect squares makes evaluating square roots easier. For instance, given the expression \( \sqrt{64} \), we recognize that 64 is a perfect square because \(8 \times 8 = 64 \). Thus, \( \sqrt{64} = 8 \). Familiarizing yourself with perfect squares up to at least 100 can save a lot of time and simplifying steps in mathematics.
simplifying expressions
Simplifying expressions is a core skill in algebra. This process involves rewriting mathematical expressions in their most compact and understandable form. When simplifying a square root, follow these general steps:

  • Identify if the number under the square root is a perfect square. If it is, find its square root directly. For example, \(\sqrt{36} = 6 \) because \(\ 6 \times 6 = 36 \).
  • If the number is not a perfect square, try to factor it into a product of perfect squares and simplify each part individually. For example: \( \sqrt{50} \ = \ \sqrt{25 \times 2} = \ 5 \sqrt{2} \). Here, 50 is broken down into 25 (a perfect square) and 2.
Sometimes, simplifying expressions involves additional operations like adding or subtracting square roots. For instance, if you need to simplify \( \sqrt{9} \ + \sqrt{16} \), it becomes \( \ 3 \ + \ 4 = \ 7\.\).Remember, practice is key. Try different problems to become more comfortable with these steps.

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Most popular questions from this chapter

Reduce to lowest terms. $$\frac{6+\sqrt{12}}{8}$$

Rationalize the denominators and simplify. $$\frac{8}{\sqrt{5}-\sqrt{3}}-\frac{12}{\sqrt{3}}$$

Rationalize the denominators and simplify. $$\frac{12}{\sqrt{15}+\sqrt{3}}-\frac{30}{\sqrt{15}}$$

Solve the following problems algebraically. Be sure to label what the variable represents. Xavier made three investments at \(6.5 \%, 7.6 \%,\) and \(9.2 \% .\) The amount invested at \(7.6 \%\) is \(\$ 1000\) less than the amount invested at \(9.2 \%,\) and the amount invested at \(6.5 \%\) is twice the amount invested at \(7.6 \% .\) If the annual income from the three investments is \(\$ 837,\) how much is invested altogether?

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$8 \sqrt{25}-3 \sqrt{21}$$
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