Question

Simplify the expression: $\sqrt[4]{{81}} + 2\sqrt[5]{{32}} - 8\sqrt {225} + \sqrt[3]{{216}}$.

Answer 1

Hint: According to given in the question we have to solve the expression $\sqrt[4]{{81}} + 2\sqrt[5]{{32}} - 8\sqrt {225} + \sqrt[3]{{216}}$ with the help of the formula given below:

Formula used:
$\sqrt {a \times a} = a$ or $\sqrt {{a^2}} = a$……………….(1)
So, with the help of the formula given (1) above we can find the square root of the term given in the expression.
$\sqrt[3]{{a \times a \times a}} = a$ or $\sqrt[3]{{{a^3}}} = a$………….(2)
With the help of the formula (2) given above we can find the cube root of the term given in the expression.
$\sqrt[4]{{a \times a \times a \times a}} = a$or $\sqrt[4]{{{a^4}}} = a$…………………(3)
So, with the help of the formula (3) given above we can find the root of the term given in the expression.
$\sqrt[5]{{a \times a \times a \times a \times a}} = a$ or $\sqrt[5]{{{a^5}}} = a$…………….(4)
With the help of the formula () given above we can find the cube root of the term given in the expression.
Now, after solving all the terms of the given expression we can find it’s value.

Complete step by step answer:
Given expression: $\sqrt[4]{{81}} + 2\sqrt[5]{{32}} - 8\sqrt {225} + \sqrt[3]{{216}}$………………(5)
Step 1: First of all we have to solve the terms of the given expression separately with the help of the formulas as mentioned in the solution hint. So, we will solve the first term which is $\sqrt[4]{{81}}$ with the help of the formula (3) as mentioned in the solution hint.
$
   \Rightarrow \sqrt[4]{{81}} = \sqrt[4]{{3 \times 3 \times 3 \times 3}} \\
   \Rightarrow \sqrt[4]{{81}} = 3 \\
 $
Step 2: Now, we have to solve the second term which is with the help of the formula (4) as mentioned in the solution hint.
$
   \Rightarrow 2\sqrt[5]{{32}} = 2\sqrt[5]{{2 \times 2 \times 2 \times 2 \times 2}} \\
   \Rightarrow 2\sqrt[5]{{32}} = 2 \times 2 \\
   \Rightarrow 2\sqrt[5]{{32}} = 4 \\
 $
Step 3: Now, we have to solve the third term which is $8\sqrt {225} $ with the help of the formula (1) as mentioned in the solution hint.
$
   \Rightarrow 8\sqrt {225} = 8\sqrt {15 \times 15} \\
   \Rightarrow 8\sqrt {225} = 8 \times 15 \\
   \Rightarrow 8\sqrt {225} = 120 \\
    \\
 $
Step 4: Same as, we have to solve the third term which is $8\sqrt {225} $ with the help of the formula (1) as mentioned in the solution hint.
$
   \Rightarrow \sqrt[3]{{216}} = \sqrt[3]{{6 \times 6 \times 6}} \\
   \Rightarrow \sqrt[3]{{216}} = 6 \\
 $
Step 5: Now, we have to substitute all the values of the terms as obtained in the step 1, 2, 3, and 4 in the given expression (1) to obtain the value of the expression.
$
   = 3 + 4 - 120 + 6 \\
   = 13 - 120 \\
   = - 107 \\
 $

Hence, with the help of the formulas as mentioned in the solution hint we have obtained the value of the given expression: $\sqrt[4]{{81}} + 2\sqrt[5]{{32}} - 8\sqrt {225} + \sqrt[3]{{216}}= - 107$

Note:
It is easy to solve expressions by finding the values of all it’s parts separately and after finding the values of all the terms we can substitute the terms in the expression to solve it.
To solve the given expression we must follow the BODMAS (Brackets, Divide, Multiply, Addition, Subtraction) rule to find the correct value of the given expression.