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Chapter 8: Problem 52

What is the difference between \(9^{2}\) and \(\sqrt{9}\) ?

Short Answer

Expert verified
78

Step by step solution

01

- Calculate 9 squared

First, calculate the value of 9 squared. \[9^2 = 9 \times 9 = 81\]
02

- Calculate the square root of 9

Next, find the square root of 9. \[\sqrt{9} = 3\]
03

- Find the difference

Subtract the square root of 9 from 9 squared. \[81 - 3 = 78\]

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Key Concepts

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

Exponents
Understanding exponents is essential in algebra. An exponent tells us how many times to multiply a number by itself. For example, in the expression \(9^2\), the number 9 is the base, and 2 is the exponent. This means you multiply 9 by itself: \[9 \times 9 = 81\]. Exponents are used in various mathematical calculations, helping to simplify and manage large numbers.
Square Roots
Square roots are a crucial concept in fundamental algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \(3 \times 3 = 9\). The square root symbol is \(\sqrt{} \), which makes the operation easy to identify. Square roots help in solving quadratic equations and in understanding the properties of numbers.
Subtraction
Subtraction is one of the basic arithmetic operations, symbolized by a minus sign ( - ). It represents the process of taking one quantity away from another. In this exercise, we subtract the square root of 9 from 9 squared: \[81 - 3\]. The result is 78. Subtraction is frequently used in various mathematical problems, including algebra, to determine the difference between numbers.
Basic Algebra Operations
Basic algebra operations include addition, subtraction, multiplication, and division of algebraic expressions. These operations follow the rules of arithmetic but are applied to variables and constants. In our exercise, we use both exponents and subtraction, showcasing different algebraic operations working together. Understanding these basic operations is vital for progressing to more complex algebra and problem-solving.

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Most popular questions from this chapter

Simplify. (a) \(625^{\frac{1}{4}}\) (b) \(243^{\frac{1}{5}}\) (c) \(32^{\frac{1}{5}}\)

Explain what is meant by the \(n^{t h}\) root of a number.

Simplify. Assume all variables are positive (a) \(\frac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) \(\left(\frac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}\)

Simplify using absolute values as necessary. (a) \(\sqrt{196 a^{2} b^{2}}\)(b) \(\sqrt{81 p^{24} q^{6}}\)(c) \(\sqrt[3]{27 p^{45} q^{9}}\)

Simplify. (a) \(\sqrt{45}+\sqrt{80}\)(b) \(\sqrt[3]{81}-\sqrt[3]{192} (c) \frac{5}{2} \sqrt[4]{80}+\frac{7}{3} \sqrt[4]{405}\)
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