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Chapter 8: Problem 37

Simplify the squares and square roots. $$\sqrt{81}$$

Short Answer

Expert verified
9

Step by step solution

01

Identify the Radicand

Recognize that the number inside the square root, known as the radicand, is 81.
02

Determine the Square Root

Find the number that, when multiplied by itself, gives 81. Identify that this number is 9 because \(9 \times 9 = 81\).

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Key Concepts

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

Radicand
In this exercise, the radicand is the number inside the square root symbol. For \( \sqrt{81} \), the radicand is 81. Understanding the radicand is crucial because it tells us what number we're dealing with before simplifying the square root. Keep the following points in mind:
  • The radicand is always inside the \( \sqrt{} \) symbol.
  • It can be any non-negative number.
  • Knowing the radicand helps you anticipate the next steps in solving the problem.
So for \( \sqrt{81} \), the radicand is 81, which is the number we need to simplify.
Square Root
The square root essentially asks, 'What number, when multiplied by itself, equals the radicand?' For \( \sqrt{81} \), we need to find a number that, when squared, equals 81. The calculation is:
\[ 9 \times 9 = 81 \] So, the square root of 81 is 9. Let's break it down further:
  • The square root of a number \( x \) seeks a number that produces \( x \) when squared.
  • The square root of 81 can be written as both \( \sqrt{81} \) and as 9.
  • Recognizing perfect squares, like 81, makes solving easier.
This makes 9 the simplest form of \( \sqrt{81} \).
Multiplication
Understanding multiplication is key to simplifying square roots. We used the fact that \( 9 \times 9 \) equals 81 to find the square root of 81. Here's why multiplication is important:
  • It determines if a number is a perfect square.
  • Helps verify the correctness of your square root.
  • Can be used to break down more complex radicands.
When simplifying square roots, knowing your multiplication facts is invaluable. Always remember:
  • If \( a \times a = b \), then \( \sqrt{b} = a \).
  • Practice breaking down numbers with multiplication to understand their square roots better.
This way, you can confidently approach simplifying square roots, using multiplication as your tool.

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