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Chapter 8: Problem 4

Quickly solve these square root problems. \(\sqrt{81}\)

Short Answer

Expert verified
The square root of 81 is 9.

Step by step solution

01

Understand the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\text{if } a^2 = b\), then \(a\) is the square root of \(b\).
02

Identify the Given Number

In this exercise, the number given is 81 and we need to find its square root.
03

Find the Square Root

Determine which number, when multiplied by itself, results in 81. We know that \(9 \times 9 = 81\). Therefore, the square root of 81 is 9.

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Key Concepts

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

square roots
A square root of a number is a value that, when multiplied by itself, gives the original number. It is one of the most fundamental operations in algebra. For example, the square root of 81 is 9 because multiplying 9 by itself results in 81.
  • Notationally, we represent it as \(\text{鈭殅(b)\), meaning 'the square root of b'.
  • If \(a^2 = b\), then \(a\) is the square root of \(b\).
Understanding square roots helps in various math problems, especially in solving quadratic equations and simplifying expressions.

To make it simpler:
  • Always look for a number which repeated multiplication (squared) matches the given number.
  • Remember that every positive number has two square roots, one positive and one negative, e.g., \(\text{卤9}\).
multiplication
Multiplication is a fundamental arithmetic operation that is essentially repeated addition. For example, \(9 \times 9\) means adding 9 to itself 9 times.
  • Consider the expression \(a \times b\). It represents the total sum when \(a\) is added to itself \(b\) times.
In the context of square roots and squares:
  • Knowing multiplication helps in finding square roots since squaring a number is the same as multiplying the number by itself.
  • For instance, determining that \(\text{鈭殅(81) = 9\) involves recognizing that \(9 \times 9 = 81\).
Thus, mastering basic multiplication makes understanding square roots much easier.
exponents
Exponents represent how many times a number, called the 'base', is multiplied by itself. For instance, \(9^2\) means 9 is multiplied by itself, resulting in \(81\).
  • The expression \(a^n\) means 'a' is raised to the power of 'n'.
When applying exponents:
  • It's a shorthand way of showing repeated multiplication.
  • In square roots, the concept is inverted. For example, finding the square root of 81 is the same as asking what number raised to the power of 2 gives 81.
Essentially, understanding exponents helps demystify square roots since squaring and taking the square root are inverse operations.

Grasping these concepts solidifies your foundational math skills.

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Most popular questions from this chapter

In each case, simplify the radical expressions by placing them under the same radical sign. \((\sqrt{x})(\sqrt{x}) \quad(x>0)\)

Change the radical expressions into exponential expressions. \(\sqrt[5]{7^{2}}\)

In each case, simplify the radical expressions by placing them under the same radical sign. \((\sqrt{(12)})(\sqrt{3})\)

Change the exponential expressions into radical expressions. \(10^{\frac{1}{2}}\)

Simplify the following radical expressions by factoring. \(\sqrt{125}\)
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