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Chapter 8: Problem 4

Simplify. (a) \(\sqrt{144}\) (b) \(-\sqrt{121}\)

Short Answer

Expert verified
(a) 12 (b) -11

Step by step solution

01

- Simplify \(\sqrt{144}\)

\sqrt{144}\ is asking for the positive square root of 144. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, \(12 \times 12 = 144\), so \(\sqrt{144} = 12\).
02

- Simplify \(-\sqrt{121}\)

\-\sqrt{121} is asking for the negative square root of 121. The square root of 121 is a value that, when multiplied by itself, gives 121. In this case, \(11 \times 11 = 121\), so \sqrt{121} = 11\. Therefore, \(-\sqrt{121} = -11\).

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Key Concepts

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

Square Roots
Square roots are a fundamental concept in mathematics. They represent a value that, when multiplied by itself, gives the original number. For instance, the square root of 144 is 12 because 12 multiplied by itself (12 脳 12) equals 144. We denote square roots using the radical symbol (鈭�). When we see 鈭�144, we ask, 'What number, when squared, equals 144?' The answer is 12.
Understanding square roots helps students in various math topics, from algebra to geometry. It is crucial to remember that the square root function gives the principal or positive root unless specified otherwise.
Simplification
Simplification is the process of reducing a mathematical expression to its simplest form. For square roots, this involves finding the value that, when squared, gives the original number. For example, simplifying 鈭�144 means finding a number that squares to 144, which is 12.
To simplify a square root:
  • Factor the number into its prime factors or consider perfect squares.
  • Identify the square root of the perfect square factor.
  • Write the simplified form.
This practice makes complex problems easier to handle and helps in solving equations efficiently.
Negative Square Roots
Sometimes, you encounter negative signs in front of square roots, which indicates the negative root. For example, -鈭�121 is asking for the negative square root of 121. Here, we find that the square root of 121 is 11, so the negative square root will be -11.
Remember, placing a negative sign in front of the square root symbol does not change the process of finding the square root itself; it just changes the sign of the result. The steps remain the same: find the square root, then apply the negative sign. In mathematics, this concept is crucial for understanding more complex functions and equations.
Mathematical Operations
Mathematical operations involving square roots follow specific rules similar to other numbers. When simplifying square roots or dealing with negative square roots, ensure you apply operations correctly. For instance, while multiplying square roots like 鈭歛 脳 鈭歜 = 鈭�(ab), note the underlying property.
Additionally, operations like addition and subtraction are different. You must only combine like terms in addition or subtraction, similar to algebraic expressions. Understanding these concepts helps perform accurate calculations and simplifies solving real-world problems.

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Most popular questions from this chapter

Simplify. Assume all variables are positive (a) \(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) (b) \(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

Simplify using absolute value signs as needed. (a) \(\sqrt{150 m^{9} n^{3}}\)(b) \(\sqrt[3]{81 p^{7} q^{8}}\)(c) \(\sqrt[4]{162 c^{11} d^{12}}\)

Use the Quotient Property to simplify square roots. \(\sqrt{\frac{180 s^{10}}{144}}\)

Approximate each root and round to two decimal places. (a) \(\sqrt{53}\) (b) \(\sqrt[3]{147}\) (c) \(\sqrt[4]{452}\)

Explain why the process of finding the domain of a radical function with an even index is different from the process when the index is odd.
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