Question

Simplify : [( 1/ 2 ) ^−3 + ( 1 /3 )^ −3 + ( −2)^ 3 ] ÷ ( 1/ 3 ) ^−2


full explanation...no wrong answers​

Answer 1

Answer:

Question :

Simplify :

${\implies{\rm{\pink{\Bigg[\bigg( \dfrac{1}{2} \bigg)^{ - 3} + \bigg( \dfrac{1}{3} \bigg)^{ - 3} + ( - 2)^{3} \Bigg] \div \bigg( \dfrac{1}{3} \bigg)^{ - 2}}}}}$

$\begin{gathered}\end{gathered}$

Solution :

Using law of exponent rules to solve this question :

$\star \: \small\boxed{ \frac{a}{b}^{m} = \dfrac{a^m}{b^m}}$

$\star \: \small\boxed{ {a}^{ - n} = \dfrac{1}{a}^{n}}$

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Now, by using this law of exponents, simplifying the given question :

${\implies{\rm{\purple{\Bigg[\bigg( \dfrac{1}{2} \bigg)^{ - 3} + \bigg( \dfrac{1}{3} \bigg)^{ - 3} + ( - 2)^{3} \Bigg] \div \bigg( \dfrac{1}{3} \bigg)^{ - 2}}}}}$

${\implies{\rm{\purple{\Bigg[\bigg( \dfrac{1^{ - 3} }{2^{ - 3} } \bigg) + \bigg( \dfrac{1^{ - 3} }{3^{ - 3} } \bigg) + ( - 2)^{3} \Bigg] \div \bigg( \dfrac{1^{ - 2} }{3^{ - 2} } \bigg)}}}}$

As we know that the (1 to the power of any number is 1). So,

${\implies{\rm{\purple{\Bigg[\bigg( \dfrac{1}{2^{ - 3} } \bigg) + \bigg( \dfrac{1}{3^{ - 3} } \bigg) + ( - 2)^{3} \Bigg] \div \bigg( \dfrac{1}{3^{ - 2} } \bigg)}}}}$

${\implies{\rm{\purple{\Bigg[(2)^{3} + (3)^{3} + ( - 2)^{3} \Bigg] \div (3)^{2} }}}}$

${\implies{\rm{\purple{\Bigg[(2 \times 2 \times 2) + (3 \times 3 \times 3) + ( - 2 \times - 2 \times - 2)\Bigg] \div (3 \times 3) }}}}$

${\implies{\rm{\purple{\Bigg[(8) + (27) + ( - 8)\Bigg] \div (9)}}}}$

Now, we know that [(+)(-) = (-)]. Then,

${\implies{\rm{\purple{\Bigg[(8 + 27- 8)\Bigg] \div (9)}}}}$

${\implies{\rm{\purple{\Bigg[(35- 8)\Bigg] \div (9)}}}}$

Opening square bracket to continuously doing the question :

${\implies{\rm{\purple{27 \div 9}}}}$

${\implies{\rm{\purple{ \dfrac{27}{9} }}}}$

Cutting the fraction into simplest form to give right answer.

${\implies{\rm{\purple{ \cancel{\dfrac{27}{9} }}}}}$

${\implies{\rm{\purple{3}\qquad \big( Ans \big) }}}$

Hence, the answer is 3.

$\begin{gathered}\end{gathered}$

Learn More :

☼ EXPONENT :

↝ The exponent of a number says how many times to use the number in a multiplication.

☼ LAW OF EXPONENT :

The important laws of exponents are given below:

• ➠ ${\rm{{a}^{m} \times {a}^{n} = {a}^{m + n}}}$
• ➠ ${\rm{{a}^{m}/{a}^{n} = {a}^{m - n}}}$
• ➠ ${\rm{({a}^{m})^{n} = {a}^{mn}}}$
• ➠ ${\rm{{a}^{n}/{b}^{n} = ({a/b})^{n} }}$
• ➠ ${\rm{{a}^{0} = 1}}$
• ➠ ${\rm{{a}^{ - m} = {1/a}^{m}}}$
• ➠ ${\rm{{a}^{\frac{1}{n} } = \sqrt[n]{a}}}$

☼ Algebraic identities :

• ➛ (a+b)²+(a-b)² = 2a²+2b²
• ➛ (a+b)²-(a-b)² = 4ab
• ➛ (a+b)(a -b) = a²-b²
• ➛ (a+b+c)² = a²+b²+c²+2ab+2bc+2ca
• ➛ (a-b)³ = a³-b³-3ab(a-b)
• ➛ (a³+b³) = (a+b)(a²-ab+b²)
• ➛ a²+b² = (a+b)²-2ab
• ➛ a³-b³ = (a-b)(a²+ab +b²)
• ➛ If a + b + c = 0 then a³ + b³ + c³ = 3abc

☼ BODMAS :

↝ BODMAS rule is an acronym used to remember the order of operations to be followed while solving expressions in mathematics.

It stands for :-

• »» B - Brackets,
• »» O - Order of powers or roots,
• »» D - Division,
• »» M - Multiplication 
• »» A - Addition
• »» S - Subtraction.

↝ It means that expressions having multiple operators need to be simplified from left to right in this order only.

☼ BODMAS RULE :

↝ First, we solve brackets, then powers or roots, then division or multiplication (whatever comes first from the left side of the expression), and then at last subtraction or addition.

• ↠ Addition (+)
• ↠ Subtraction (-)
• ↠ Multiplication (×)
• ↠ Division (÷)
• ↠ Brackets ( )

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