1.3 Square Roots and Cube Roots PDF

## 1.3 Square Roots and Cube Roots ### In this section you will... - find the squares of positive and negative integers and their corresponding square roots - find the cubes of positive and negative integers and their corresponding cube roots - learn to recognise natural numbers, integers and rati...

## 1.3 Square Roots and Cube Roots ### In this section you will... - find the squares of positive and negative integers and their corresponding square roots - find the cubes of positive and negative integers and their corresponding cube roots - learn to recognise natural numbers, integers and rational numbers. ### Key words - cube root - natural numbers - rational numbers - square root ### Tip The natural numbers are the counting numbers and zero. $5^2 = 25$ This means that the square root of 25 is 5. This can be written as $\sqrt{25} = 5$. This is the only answer in the set of natural numbers. However $(-5)^2 = -5 \times -5 = 25$ This means that the integer -5 is also a square root of 25. Every positive integer has two square roots, one positive and one negative. 5 is the positive square root of 25 and -5 is the negative square root. No negative number has a square root. For example, the integer -25 has no square root because the equation $x^2 = -25$ has no solution. $5^3 = 125$ This means that the cube root of 125 is 5. This can be written as $\sqrt[3]{125} = 5$. You might think -5 is also a cube root of 125. However $(-5)^3 = -5 \times -5 \times -5 = (-5 \times -5) \times -5 = 25 \times -5 = -125$ So $\sqrt[3]{-125} = -5$ Every number, positive or negative or zero, has only one cube root. ### Worked example 1.3 Solve each equation. a) $x^2 = 64$ b) $x^3 = 64$ c) $x^3 + 64 = 0$