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Chapter 9: Problem 11

Simplify. $$ \sqrt{10,000} $$

Short Answer

Expert verified
100

Step by step solution

01

Identify the Expression

Recognize that the expression to simplify is \(\sqrt{10,000}\). Simplifying this square root involves finding a number that, when multiplied by itself, equals 10,000.
02

Find the Square Root

10,000 is a perfect square, which means it has an integer as its square root. By intuition or calculation, determine that \(100 \times 100 = 10,000\). Therefore, \(\sqrt{10,000} = 100\).

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Key Concepts

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

Square Root Calculation
Square root calculation involves finding a number that, when multiplied by itself, results in the given value. For example, consider \(\sqrt{10,000}\).
Our goal is to determine a number which, when squared (i.e., multiplied by itself), equals the number inside the square root symbol.
To calculate basic square roots, you can use intuition, arithmetic, or a calculator. In this case, you're finding \(\sqrt{10,000}\). We know that 100 multiplied by itself equals 10,000, so \(\sqrt{10,000} = 100\).
Square roots simplify many mathematical problems, making calculations easier and more manageable.
Perfect Squares
A perfect square is a number that is the square of an integer. For instance, \(100 \times 100 = 10,000\), making 10,000 a perfect square.
Recognizing perfect squares helps in simplifying square roots efficiently. Here are some tips to identify them:
  • Notice common perfect squares like 1, 4, 9, 16, 25, and so on.
  • Learn the squares of the first 10 integers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
  • Use these known values to quickly identify larger perfect squares.
Understanding and memorizing smaller perfect squares will make larger ones easier to recognize.
Basic Arithmetic
Basic arithmetic is crucial for simplifying square roots. Here鈥檚 a quick overview:
Multiplication: To simplify \(\sqrt{10,000}\), you need to know that \(100 \times 100 = 10,000\). Practicing multiplication helps speed up finding square roots.
Division: Often, dividing large numbers into smaller factors can help identify perfect squares.
Addition and Subtraction: Although not directly involved in simplifying square roots, strong skills here support overall mathematical fluency.
Practicing basic arithmetic builds a strong foundation for solving more complex problems.

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Most popular questions from this chapter

For each right triangle, find the length of the side not given. Assume that \(c\) represents the length of the hypotenuse. Give an exact answer and, when appropriate, an approximation to three decimal places. $$ a=1, c=32 $$

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