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Chapter 8: Problem 27

Find each square root. $$\sqrt{81}$$

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 value is a number which, when multiplied by itself, gives the original value. In mathematical terms, if \(\sqrt{a} = b\), then \(b^2 = a\). In this case, we are looking for the number \(b\), such that \(b^2 = 81\).
02

Find the square root

Starting to count from 1 and up, we look for a number that, when squared (which means multiplied by itself), equals to 81. If we try with 9, we can see that \(9^2 = 81\).

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Key Concepts

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

Finding square roots
Finding square roots can be a fascinating journey into understanding how numbers relate to each other. The square root of a number is essentially the value that, when multiplied by itself, gives the original number. For instance, to find the square root of 81, we are looking for a number which, when squared, equals 81.
This process involves estimating and checking through multiplication.
  • Start small, testing numbers you are comfortable multiplying.
  • Use familiar reference points; for example, since you know that 9 multiplied by 9 equals 81, then the square root of 81 is 9.

With practice and patience, finding square roots becomes more intuitive and can be done more quickly by recognizing patterns.
Perfect squares
In the realm of mathematics, **perfect squares** are numbers that are the product of an integer multiplied by itself. Knowing and recognizing perfect squares can greatly simplify the task of finding square roots.
Here are a few important aspects to understand about perfect squares:
  • Perfect squares are always non-negative since multiplying two positive or two negative numbers results in a positive product.
  • Numbers like 1, 4, 9, 16, 25, and 81 are all classic examples of perfect squares because they can be simplified to the square of an integer like 1, 2, 3, 4, 5, and 9, respectively.
Familiarizing oneself with the list of perfect squares can aid in quickly and accurately determining square roots, especially when dealing with common values in algebraic problems.
Algebra basics
Algebra lays the foundation for understanding mathematical concepts such as square roots and perfect squares. It introduces variables and equations, allowing the representation of numbers and operations in a more general form.
  • **Variables and Constants**: Symbols (like x or y) that can represent numbers within equations.
  • **Expressions and Equations**: Expressions are combinations of numbers, variables, and operations without an equality sign, while equations will include equality to solve for unknowns.
  • **Simplifying**: The process of combining like terms to reduce an expression to its simplest form.
The concepts of square roots and perfect squares fit neatly into algebra as they often involve the manipulation of expressions and solving of equations. Understanding these basics will make finding square roots more straightforward as it involves practicing the logical steps inherent in algebraic manipulation.

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Most popular questions from this chapter

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{720}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\left(\frac{64}{125}\right)^{1 / 3}$$

Solve: \(\sqrt[4]{3 x+4}=5\)

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\left(3^{2 / 3} 5^{1 / 3}\right)^{3}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{\frac{250 a^{3} b^{4}}{16 b}}$$
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