Estimating with Square Roots

Transcription

1 ACTIVITY 3.2 Estimating with Square Roots The square root of most numbers is not an integer. You can estimate the square root of a number that is not a perfect square. Begin by determining the two perfect squares closest to the radicand so that one perfect square is less than the radicand, and one perfect square is greater than the radicand. Then consider the location of the expression on a number line and use approximation to estimate the value. WORKED EXAMPLE To estimate 10 to the nearest tenth, identify the closest perfect square less than 10 and the closest perfect square greater than 10. The closest perfect square less than 10: 9 The square root you are estimating: 10 The closest perfect square greater than 10: You know: This means that the estimate of 10 is between 3 and 4. Locate each square root on a number line. The approximate location of 10 is closer to 3 than to 4 when plotted The symbol means approximately equal to Think about the location of 10 in relation to the values of 3 and 4. Therefore, Calculate the square of 3.2 to determine if it is a good estimation of 10. Adjust the estimated value if necessary. M4-36 TOPIC 1: The Real Number System

2 2. Consider each expression a. Order the expressions from least to greatest. b. Locate the approximation of each expression on the number line. Explain the strategy you used to plot each value. To locate the approximation of a square root on a number line, identify the two closest perfect squares, one greater than the radicand and one less than the radicand c. Estimate the value of each expression to the nearest tenth. Then, calculate the square of each approximation to determine if it is a good estimation. Adjust the estimated value, if necessary. 3. Solve each equation. Round your answer to the nearest tenth. a. x b. a If x 2 5 4, then x 5 4. Use this fact to show the solution to each equation. c. c d. g LESSON 3: What Are Those?! M4-37

3 ACTIVITY 3.3 Cubes and Cube Roots In the previous activity, you investigated squares and square roots. Now, let s consider cubes and cube roots. 1. Use unit cubes to build three different cubes with the given side lengths. Then complete the table. a. 1 unit b. 2 units c. 3 units Dimensions of Each Cube Total Number of Unit Cubes The formula for the volume of a cube is V 5 s 3 s 3 s, which can be written as V 5 s 3. You just calculated the volume of 3 cubes whose side lengths were the first 3 counting numbers, , , and The numbers 1, 8, and 27 are called perfect cubes. A perfect cube is the cube of a whole number. For example, 64 is a perfect cube since 4 is a whole number, and To calculate the cube of a number, you multiply the number by itself 3 times. M4-38 TOPIC 1: The Real Number System

4 2. Calculate the cubes of the first 10 whole numbers If you know the volume of a cube, you can work backwards to calculate the side lengths of the cube. For example, to determine the side lengths of a cube that has a volume of 125, you need to determine what number used as a factor 3 times will equal 125. Since , a side length of the cube is 5, and 5 is called the cube root of 125. A cube root is one of 3 equal factors of a number. As with the square root, the cube root also uses a radical symbol but has a 3 as an index: 3 1. The index is the number placed above and to the left of the radical to indicate what root is being calculated. The cube root of a number that is not a perfect cube is often an irrational number. 3. Write the cube root for each perfect cube _ What is the side length of the largest cube you can create with 729 cubes? 5. Will the cube root of a number always be a whole number? If not, provide an example of a cube root that is not an integer. LESSON 3: What Are Those?! M4-39

5 Remember, the radicand is under the. Most numbers do not have whole numbers for their cube root. Let s estimate the cube root of a number using the same method used to estimate the square root of a number. WORKED EXAMPLE To estimate 3 33 to the nearest tenth, first identify the two perfect cubes closest to the radicand. One of the perfect cubes must be less than the radicand, and the other must be greater than the radicand. Then consider the location of the expression on a number line and use approximation to estimate the value. The closest perfect The cube root The closest perfect cube less than 33: you are estimating: cube greater than 33: You know: This means that the estimate of 3 33 is between 3 and 4. Locate the approximate value of 3 33 on a number line Next, choose decimals between 3 and 4, and calculate the cube of each decimal to determine which one is the best estimate. Consider: (3.2)(3.2)(3.2) (3.3)(3.3)(3.3) Therefore, M4-40 TOPIC 1: The Real Number System

6 6. Identify the two closest perfect cubes, one greater than the radicand and one less than the radicand. Then locate the approximation of each expression on a number line. Finally, estimate each cube root to the nearest tenth. a NOTES b c Solve each equation. Round to the nearest tenth. a. x b. a c. c LESSON 3: What Are Those?! M4-41

7 TALK the TALK % 20% Venn Diagrams and Real Numbers Combining the set of rational numbers and the set of irrational numbers produces the set of real numbers. You can use a Venn diagram to represent how the sets within the set of real numbers are related. 1. The Venn diagram shows the relationship between the six sets of numbers shown. Write each of the 30 numbers in the appropriate section of the Venn diagram. Real Numbers % 23 p ,513 1,000, Rational Numbers Integers Whole Numbers Irrational Numbers Natural Numbers M4-42 TOPIC 1: The Real Number System

8 NOTES 2. Use your Venn diagram to decide whether each statement is true or false. Explain your reasoning. a. A whole number is sometimes an irrational number. b. A real number is sometimes a rational number. c. A whole number is always an integer. LESSON 3: What Are Those?! M4-43