Mathematics: Squares and Cubes

Mathematics: Squares and Cubes You should now try to get used to writing numbers in mathematical notation. This lesson will discuss how we can calculate squares, cube and square root andcube roots of numbers. 3×3 = 9, we all know that. Let’s write this in mathematical notation. In mathematical notation, 3×3= 3 2 3 2 The 2 is the “exponent” of the original number, 3. What the exponent does isthat it tells us how many times the original number multiplies itself by. Thenumber of times here is 2, so 3 2 = 3×3. We call this the square of a number. How about, 2 3 ? In this case, the exponent is 3, so the number 2 will multiply itself 3 times, so2 3 = 2x2x2= 8 We call this the cube of a number. And so on. 2 4 =2x2x2x=16 2 5 =2x2x2x2x2=32 …. Exercise: 1) 2 2 2) 4 2 3) 7 2

4) (2+3) 2 5) 6 2 6) 1 3 7) 4 3 8) 2 3 9) 7 3 10) 10 3 Answers: 1) 2×2=4 2) 4×4= 16 3) 7×7= 49 4) (2+3) 2 = (5) 2 = 5×5= 25 5) 6×6= 36 6) 1 × 1 × 1 = 1 7) 4 × 4 × 4 = 64 8) 2 × 2 × 2 = 8 9) 7 × 7 × 7 = 343 10) 10 × 10 × 10 = 1000 Now, let’s try undoing the process. Let’s take a number 16. We want to get two identical numbers to multiply to 16. Let’s take the two numbers to be : a And we want a x a= 16

a x a = a 2 , as we discussed just now, so: a 2 =16 What does “a” equal to? There are two ways to finding “a”. 1. Trial and Error2. Use a calculate, and the sqrt button : 16. We will use method 1 here. Now, let’s try to visualize this process here. We know 16 is between 0-100. 0 2 = 0 and 10 2 = 100 So we know that “a” is between 1 and 9. Let’s try each individual value between 1 and 9 1 2 = 1×1= 1 (Wrong) 2 2 =2×2= 4 (Wrong, but closer) 3 2 = 3×3 = 9 (Wrong, but even closer) 4 2 = 4×4 = 16 (Correct!) Since “a” is a whole number, 16 is a squared number. If “a” is an irrational number with loads of decimals, the number “a” multipliesout to is not a squared number. Exercise: Find the value of “A”. 1) A 2 =25 2) A 2 =49 3) A 2 =100 4) A 2 =121

5) A 2 =64 Answers: Use trial and error for these problems. 1) 5, 5×5= 25 2) 7, 7×7=49 3) 10, 10×10=100 4) 11,11×11=121 5) 8, 8×8=64