Estimating Roots Word Problems

Concept

Square root of a number is an inverse operation of squaring a number. The square of a number is the value of power 2 of the number, while the square root of a number is the number that we need to multiply by itself to get the original number. If ‘a’ is the square root of ‘b’, it means that a × a = b. The square of any number is always a positive number, so every number has two square roots, one positive value, and one negative value. For example, both 2 and -2 are square roots of 4.

Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. Thus, the cube for the number (x) becomes x³ or x-cubed. While on the other hand, the cube root of a number is the reverse process of the cube of a number and is denoted by ∛.

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and realistic. This method helps in estimating and approximating the square root of a given number.

Rules

1. Read the word problem and set up an equation.
2. Isolate the variable, by taking the square root/cube root of both sides of the equation.
3. Find the largest perfect square/cube less than the given value.
4. Find the smallest perfect square/cube greater than the given value.
5. Find which perfect square/cube the original value is closer to.
6. Calculate the square root/ cube root of the nearest perfect square/cube the original value is closer to.

Example

The volume of a cube is 350 cm³. Estimate the length of one side of the cube.

Solution

To find the length of one side, you have to find the cube root of 350. 350 is between the perfect cubes 343 and 512. It is closer to 343. The cube root of 343 is 7. So,

$\fn_phv \sqrt[3]{350}\approx 7$