Question

Simplify ${8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right)$.

Answer 1

Hint: Use the BODMAS rule to solve the expression. First, solve the bracket. After that find the square root and also square the first value and then find the cube root of that. Then multiply the terms. After that add the terms. Lastly, do the subtraction to get the desired result.

Complete step-by-step solution:
BODMAS is a short form for Brackets, of, Division, Multiplication, Addition, and Subtraction. In some regions, it is also known as PEDMAS - Parentheses, Exponents, Division, Multiplication, Addition, and Subtraction.

B	Brackets \[\left[ {\left\{ {\left( - \right)} \right\}} \right]\]	P (Parentheses)
O	Orders, Of	E (Exponents)
D	Division $ \div ,/$	D
M	Multiplication $*, \times , \cdot $	M
A	Addition, +	A
S	Subtraction, –	S

According to the BODMAS rule, first, remove the parentheses,
$ \Rightarrow {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \dfrac{1}{{144}}$
After that solve the exponent part,
As we know,
${a^{\dfrac{m}{n}}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}}$
Now take the ${n^{th}}$ root of the expression,
${a^{\dfrac{m}{n}}} = \sqrt[n]{{{a^m}}}$
Use this to find the value of the first term,
$ \Rightarrow {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = \sqrt[3]{{64}} - 3 \times 10 + \dfrac{1}{{144}}$
Simplify the terms,
$ \Rightarrow {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = 4 - 3 \times 10 + \dfrac{1}{{144}}$
Now multiply the terms,
$ \Rightarrow {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = 4 - 30 + \dfrac{1}{{144}}$
Now find the LCM on the right side,
$ \Rightarrow {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = \dfrac{{576 - 4320 + 1}}{{144}}$
Simplify the terms,
$\therefore {8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right) = - \dfrac{{3743}}{{144}}$

Hence, the value of ${8^{\dfrac{2}{3}}} - \sqrt 9 \times 10 + \left( {\dfrac{1}{{144}}} \right)$ is $ - \dfrac{{3743}}{{144}}$.

Note: This rule explains the order of operations (order of precedence) to solve an expression. According to the BODMAS rule, in a given mathematical expression that contains a combination of signs: brackets ((), {}, [], −), multiplication, of, addition, subtraction, division, we must first solve or simplify the brackets followed by of (powers and roots, etc.), of, then division, multiplication, addition and subtraction from left to right. Solving the problem without following the BODMAS rule or from the right-hand direction will give you a wrong answer.