The square root of a number p is a number whose square is equal to p. Therefore, a square root of a number p is a solution to the equation x²=p. Every positive number has a positive and a negative square root. A perfect square is a number with integers as its square roots.
The symbol √ is called a radical sign. It is used to represent a square root. The number under the radical sign is called the radicand. For example, √25 = 5 and √12.25 = 3.5.
When solving an equation by taking square roots, it is important to consider both the positive and the negative square roots. For instance, when solving x² = 81, the solution is x = ± 9, resulting in x = -9 and x = 9.
The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse, represented by the formula a² + b² = c².
For example, in a right triangle with legs of length 5 and 12, the hypotenuse length can be calculated as follows: 5² + 12² = c², which simplifies to 25 + 144 = c², resulting in 169 = c² and c = 13.
Another example is a right triangle with legs of length 9 and 2.1, which would result in 9² + 2.1² = 2.9², leading to a = 2 as the solution.
A cube root of a number p is a number whose cube is equal to p. So, a cube root of a number p is a solution to the equation x³ = p. The symbol &√ is used to represent a cube root. When solving equations of the form x³ = p, it is important to consider the inverse operation and take the cube root of each side.
For example, to find the cube root of 216, the solution is √√√x³ = √√216, resulting in x = 6. Similarly, for the equation n³ = -8, the cube root can be calculated as √√√³² = ²√√-8.
A rational number is a number that can be written as a fraction where both a and b are integers and b ≠ 0. Rational numbers can be written as either terminating decimals or repeating decimals. For example, 12/15 can be written as 0.8, and 5/33 can be written as 0.15 repeating.
An irrational number is a number that cannot be expressed as a fraction where a and b are integers and b ≠ 0. The square root of any whole number that is not a perfect square and the cube root of any integer that is not a perfect cube are examples of irrational numbers. Additionally, the decimal form of an irrational number neither terminates nor repeats, such as √2 and √3.
If the equation a² + b³²=c² is true for the side lengths of a triangle, then the triangle is a right triangle. The converse of the Pythagorean Theorem states that when the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, it forms a right triangle. This can be used to test whether a triangle is a right triangle.
