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Chapter 5: Problem 23

Simplify. $$ \sqrt{162} $$

Short Answer

Expert verified
Simplified form is \( 9 \sqrt{2} \).

Step by step solution

01

Identify perfect square factors

First, identify the factors of 162 that include a perfect square. Notice that 162 can be factored as 81 and 2, and 81 is a perfect square.
02

Express inside of radical

Rewrite the square root of 162 using its factors: \ \( \sqrt{162} = \sqrt{81 \cdot 2} \ \)
03

Simplify the square root

Simplify using the property of square roots \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) : \ \( \sqrt{81 \cdot 2} = \sqrt{81} \cdot \sqrt{2} \ \)
04

Calculate the square root of the perfect square

Since 81 is a perfect square and \( \sqrt{81} = 9 \) : \ \( \sqrt{81} \cdot \sqrt{2} = 9\sqrt{2} \ \)

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Key Concepts

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

Perfect Square Factors
To simplify square roots, it's important to identify perfect square factors within the number under the radical. A perfect square is an integer that is the square of another integer, like 4, 9, 16, and 81. When we break down a number like 162, we look for factors that include these perfect squares. For 162, we can factor it into 81 and 2, where 81 is a perfect square because it is equal to 9 squared. This step is crucial for simplifying because it allows us to separate the number into a part that can be simplified and another that might remain under the radical.
Properties of Square Roots
After identifying the perfect square factors, we use properties of square roots to simplify the expression. One key property is that the square root of a product is equal to the product of the square roots: \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \] Applying this, we rewrite \ \( \sqrt{162} = \sqrt{81 \cdot 2} \ \). Next, we separate the factors under the radical: \ \( \sqrt{81} \cdot \sqrt{2} \ \). This property helps simplify by isolating the square root of the perfect square, which can be easily calculated.
Radical Expressions
A radical expression involves a root symbol, usually the square root. Simplifying such expressions is easier when you know the properties of radicals and can identify perfect squares. For \ \( \sqrt{162} \ \) , we saw that 162 can be written as 81 times 2. Since \ \( \sqrt{81} \ = \ 9 \ \), \ \( \9 \sqrt{2} \ \) is the simplified form. The goal is to make the radical simpler while keeping it mathematically equivalent. In general, simplifying radicals makes calculations easier and expressions clearer.

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Most popular questions from this chapter

Similar to lines of symmetry for two-dimensional figures, three- dimensional objects can have planes of symmetry. For example, a cube has nine planes of symmetry, including these three: a. How many planes of symmetry does a regular square pyramid have? Describe or sketch them. b. How many planes of symmetry does a sphere have? Describe or sketch them. c. How many planes of symmetry does a right hexagonal prism have? Describe or sketch them.

If the angle of rotation for a figure with rotation symmetry is an integer, it is also a factor of 360. Consider what might happen if you tried to create a figure using an angle measure that isn鈥檛 a factor of 360, such as 135掳. Choose a point A and a center of rotation. Rotate Point A 135掳, and rotate the image 135掳. Keep rotating the images until you return to the original point. (When you perform the rotations, you will pass the original point, because you have made one full turn.) a. How many full circles did you make? b. How many copies of the point do you have in your drawing? c. There is an angle of rotation smaller than 135掳 that you could have used to create this same design. What is its measure? d. Now find the greatest common factor (GCF) of 135 and 360. e. Divide 135 and 360 by your answer to Part d. f. Compare your answers for Parts a鈥揷 to your answers for Parts d and e. What do you notice? g. Suppose you created a figure by rotating a basic design element 80掳 each time. What angle of rotation will the final design have? Test your answer by rotating a single point.

Do your work for Exercises 6 and 7 without using tracing paper. Draw a quadrilateral \(A B C D .\) Choose a point to be the center of rotation, and rotate your quadrilateral 80掳 about that point. Use prime notation to label the image vertices. For example, the image of vertex \(A\) will be vertex \(A^{\prime}.\)

Perspective drawings look three-dimensional. The projection method for making scale drawings is related to a method for making perspective drawings. On your own paper, follow the steps below to make a perspective drawing of a box. Use a pencil. a. Start by drawing a rectangle. This will be the front of your box. b. Choose a point outside your rectangle. This point is called the vanishing point for your drawing. Connect each vertex to that point, and then find the midpoint of each connecting segment. c. Connect the four midpoints you found in Part b to each other, in order. This gives you the back of the box. Then erase the lines connecting them to the vanishing point. d. To make the box clearer, erase the lines that should be hidden on the back of the box, or make them dashed. e. Follow the same steps to make a perspective drawing of a triangular prism. That is, start with a triangle (instead of a rectangle) and follow Part a鈥揹. f. In this method of three-dimensional drawing, at what step do you create a pair of similar figures? Explain.

Evaluate each expression for a 2 and b 3. $$ \left(\frac{b}{a} \cdot \frac{a}{b}\right)^{a} $$
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