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Chapter 10: Problem 42

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$(\sqrt{100-36})^{2}$$

Short Answer

Expert verified
The value is 64.

Step by step solution

01

Inside the Square Root

First, simplify the expression inside the square root. Calculate the difference between 100 and 36: 100 - 36 = 64.
02

Simplify the Square Root

Next, take the square root of 64. Since 64 is a perfect square, its square root is 8: \[ \sqrt{64} = 8. \].
03

Apply the Squaring Operation

Finally, square the result of the square root. Since the square root of 64 is 8, calculate: \[ 8^{2} = 64. \].

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Key Concepts

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

Understanding Square Roots
Square roots are essential in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 64 is 8 because \(8 \times 8 = 64\). When you see the square root symbol (\( \sqrt{} \)), it means you are looking for a number that squares back to the given value.

It鈥檚 important to recognize 'perfect squares.' Perfect squares are numbers like 1, 4, 9, 16, and 25 that are products of a number multiplied by itself. Knowing these can help you quickly find square roots.
  • The square root of 1 is 1 (\(1^2 = 1\))
  • The square root of 4 is 2 (\(2^2=4\))
  • The square root of 9 is 3 (\(3^2=9\))
  • The square root of 64 is 8 (\(8^2=64\))
Perfect Squares Explained
Perfect squares come up often in algebra, especially when simplifying expressions. A number is a perfect square if it can be expressed as the square of an integer.
For example:
- 36 is a perfect square because \( 6 \times 6 = 36 \).
- 49 is a perfect square because \( 7 \times 7 = 49 \). These numbers are useful because they allow us to simplify square roots easily.
Recognizing perfect squares can significantly speed up your calculations. Instead of doing long divisions or multiplications, you immediately know the square root of 100 is 10 since \(10 \times 10 = 100\). Therefore, being quick with identifying these special numbers can be very beneficial!
Order of Operations
Order of operations is a fundamental concept in mathematics. It tells us the correct sequence to follow when performing calculations. The common acronym to remember this is PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Let's apply PEMDAS to the given problem \( (\sqrt{100-36})^{2} \).
First, evaluate inside the parentheses:
\[ 100 - 36 = 64 \]

Next, handle the square root:
\[ \sqrt{64} = 8 \]

Finally, apply the exponent:
\[ 8^2 = 64 \]
Following these steps ensures that we simplify and solve expressions accurately and efficiently.

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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. $$\frac{6}{\sqrt{x}+\sqrt{3}}$$

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. $$8 \sqrt{25}-3 \sqrt{21}$$

Rationalize the denominators and simplify. $$(4-\sqrt{7})^{2}-\frac{18}{4+\sqrt{7}}$$

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. $$8 \sqrt{25}-3 \sqrt{21}$$

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