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Chapter 13: Problem 8

Simplify each expression. \(\sqrt{81}\)

Short Answer

Expert verified
The simplified expression is 9.

Step by step solution

01

Identify the Radicand

The expression is \(\sqrt{81}\), where 81 is the radicand. We need to find the square root of 81.
02

Determine the Square Root

To simplify \(\sqrt{81}\), consider that 81 is a perfect square. Identify the number that when multiplied by itself gives 81, which is 9, because \(9 \times 9 = 81\).
03

Write the Simplified Expression

Since 81 is a perfect square and its square root is 9, we can simplify the expression \(\sqrt{81}\) to just 9.

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Key Concepts

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

Perfect Squares
Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. These numbers play a crucial role in simplifying square roots. In the case of \(\sqrt{81}\), we see that 81 is a perfect square. This is because 9 times 9 equals 81, or more formally, \(9 \times 9 = 81\). Identifying a number as a perfect square allows us to take its square root more easily, since it will result in another whole number.

Here are some tips to identify perfect squares:
  • They are always positive.
  • Results from squaring an integer.
  • Eliminates the need to deal with decimals or fractions when simplifying the square root.
Recognizing perfect squares lets you simplify square root expressions efficiently.
Radicand
The term radicand refers to the number inside the square root symbol (\(\sqrt{}\)). In the expression \(\sqrt{81}\), the radicand is 81.

The radicand is what we are determining the square root of; understanding this component helps in effectively applying methods to simplify the root.
  • A radicand can be any real number, but perfect squares will simplify perfectly.
  • If not a perfect square, you might end up with a decimal or fraction upon simplification.
  • Recognizing and breaking down complex radicands into perfect squares can aid simplification.
By clearly identifying the radicand, you can choose the right approach to find its square root.
Square Root Property
The square root property is a fundamental concept to understand when simplifying square roots. It states that the square root of a perfect square number will always be an integer. In the example of \(\sqrt{81}\), since 81 is a perfect square, its square root is the integer 9.

This property can be extremely useful:
  • Allows exact simplifications of expressions with perfect square radicands.
  • Keeps calculations clean and straightforward, avoiding irrational numbers.
  • Reminds us that the square root of non-perfect squares will yield a non-integer result.
Utilizing the square root property helps in recognizing numbers that are easily simplified, thus enhancing problem-solving efficiency.

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Most popular questions from this chapter

Sierra is flying a kite. She has let out 55 feet of string. If the angle of elevation is \(35^{\circ}\) and the hand holding the string is 6 feet from the ground, what is the altitude of the kite? Round to the nearest tenth.

Determine whether it is possible for a trapezoid to have the following conditions. Write yes or no. If yes, draw the trapezoid. three obtuse angles

Verify each step in parts a through e. Then solve parts f and g. a. \(\sin P=\frac{p}{q}\) and \(\cos P=\frac{r}{q}\) b. \(\sin ^{2} P=\frac{p^{2}}{q^{2}}\) and \(\cos ^{2} P=\frac{r^{2}}{q^{2}}\) C. \(\sin ^{2} P+\cos ^{2} P=\frac{p^{2}}{q^{2}}+\frac{r^{2}}{q^{2}}\) or \(\frac{p^{2}+r^{2}}{q^{2}}\) d. \(p^{2}+r^{2}=q^{2}\) e. \(\sin ^{2} P+\cos ^{2} P=\frac{q^{2}}{q^{2}}\) or 1 f. Find \(\sin x\) if \(\cos x=\frac{3}{5}\). g. Find \(\cos x\) if \(\sin x=\frac{5}{13}\).

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