Square Roots & Cube Roots

Review Calculating Exponents Exponents tell you how many times you multiply a number by itself. For example, in 42, the 2 is the exponent, which tells you to multiply 4 two times. So, 42 = 4 × 4 = 16. You can read 42 as “four to the power of two” or as “four squared.” It is called “four squared” because you multiply the side length by itself to find the area of a square. Similarly, you can read 43 as “four to the power of three” or as “four cubed.” It is called “four cubed” because you multiply the edge length three times to find the volume of a cube. 43 = 4 × 4 × 4 = 64. Try this one yourself: Evaluate 32 and 33.

Frame Paintings Amari has three square paintings. One painting has area 400 cm2. How long is each side of the frame? Let n be the length of one side. The area is n x n or n2. So n2 = 400. What number times itself is equal to 400? 20 × 20 = 400. Each side is 20 cm long. This is a square root. The square root of a number is a number that, when multiplied by itself, gives the original number. The square root of 400 is 20. You can show that with a symbol: √400 = 20. To find the square root, ask what number times itself gives you the number you started with. Amari’s second painting has an area of 81 square inches. 9 × 9 = 81, so √81 = 9. You can also use a calculator to find a square root. Type in the number you want to find the square root of, and then press the square root key. Try this one yourself: Amari’s third painting has an area of 36 square inches. How long is each side of this frame?

Plant a Garden Emily wants to plant tomato seeds in her garden. The seeds should be spread out over an area of 45 m2. If she makes a square garden, how long should the sides be? What is the square root of 45? 6 × 6 = 36, which is too low. 7 × 7 is 49, which is too high. The square root of 45 is an irrational number which means that it is a decimal that never ends and can’t be expressed as a fraction. Try it on a calculator: type 45 and then the square root button. A calculator shows that the answer is 6.70820393249, but the decimal goes on forever. When you have an irrational number as an answer, you can round it. √45 ≈ 6.7. This means that the square root of 45 is approximately equal to 6.7. The squiggly equal sign means that the answer is an approximation. Try this one yourself: What is the square root of 67? Round to the nearest tenth.

Transport an Alligator A biologist transports an alligator in a crate in the shape of a cube. The crate has a volume of 27 m3. How wide is the cube? Use your knowledge of volume. A cube is the same length on all sides. The formula for volume is length × width × height. So, what number times itself 3 times is 27? 3 × 3 = 9, 9 × 3 = 27. 3 is the cube root of 27. A cube root is a number that when multiplied by itself 3 times gives the original number. You can show this using the cube root symbol: ∛27 = 3. So, the crate measures 3 meters on each side. You can also solve cube roots on a calculator. What is the cube root of 1,728? 10 × 10 × 10 is 1,000, so it must be more than 10. Try 11 on a calculator. 11 × 11 × 11 = 1,331, which is too low. Try 12. 12 × 12 × 12 = 1,728. The cube root of 1,728 is 12. Try this one yourself: What is the cube root of 729?

Make Resin Art You have 800 cm3 of resin. You make a cube out of all the resin. What is the side length of the cube? You need to find the cube root of 800. What number times itself is equal to 800? 10 × 10 × 10 = 1,000, which is too high. 9 × 9 × 9 = 729, which is too low. Try using a calculator. Press 800, then the cube root key. The answer is 9.28317766723 and beyond. It is an irrational number. You can round the answer. ∛800 ≈ 9.3. The side length of the cube is about 9.3 cm. Try this one yourself: You have 500 cm3 of resin. You make another cube out of this resin. What is the side length of the cube? Round to the nearest centimeter.