Question

Question:- 1

three numbers are to one another 2:3:4. the sum of their cubes is 33957 find the numbers.




Question:- 2

Find the cube root of 91125.​

Answer 1

$\large\underline{ \underline{ \sf \maltese{ \: Question:- \: 1}}}$

• Three numbers are to one another 2:3:4. the sum of their cubes is 33957. Find the numbers.

$\large\underline{ \underline{ \sf \maltese{ \: Given:- }}}$

• Let the First Number be = $\sf\green{ 2x}$
• Let the Second Number be = $\sf\green{ 3x}$
• Let the Third Number be = $\sf\green{ 4x}$

$\large\underline{ \underline{ \sf \maltese{ \: Solution:- }}}$

$\begin{gathered}\begin{gathered}\begin{gathered}: \implies \underline\blue{ \boxed{\displaystyle \sf \bold\orange{\: ( \: 2x \: )^{3} \: + \: ( \: 3x \: )^{3} \: + \: ( \: 4x \: )^{3} \: = \: 33957 }} }\\ \\\end{gathered}\end{gathered}\end{gathered}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ ( \: 2x \: )^{3} \: + \: ( \: 3x \: )^{3} \: + \: ( \: 4x \: )^{3} \: = \: 33957 }}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ 8x ^{3} \: + \: 27x ^{3} \: + \: 64x ^{3} \: = \: 33957}}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ 99x ^{3} \: = \: 33957}}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ x ^{3} \: = \: \cancel\frac{33957}{99} }}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ x ^{3} \: = \: 343}}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ x \: = \: \sqrt[3]{343} }}$

$\qquad\quad {:} \longrightarrow \underline \red{\boxed{\sf{ \:x \: = \: 7 }}}$

• First Number = 2x = 2 × 7 = 14
• Second Number = 3x = 3 × 7 = 21
• Third Number = 4x = 4 × 7 = 28

$\bf \therefore \; First \;number = 14$

$\bf \therefore \; Second \; Number = 21$

$\bf \therefore \; Third \;number = 28$

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$\large\underline{ \underline{ \sf \maltese{ \: Question:- \: 2}}}$

• Find the cube root of 91125.

$\large\underline{ \underline{ \sf \maltese{ \: Solution:- }}}$

Resolving the given numbers into prime factors

• 91125 = 5 × 5 × 5 × 3 × 3 × 3 × 3 × 3 × 3

Grouping the factor in triples of equal factors

• 91125 = $\sf\green{5 \times 5 \times 5 \: } \times \sf\blue{ \: 3 \times 3 \times 3} \times \sf\red{ \: 3 \times 3 \times 3}$

Taking one factor from each triple

• $\sqrt[3]{91125} \: = \: 5 \: \times \: 3 \: \times \: 3$

$\begin{gathered}\begin{gathered}\qquad \therefore\: \sf{ \sqrt[3]{91125} \: = \underline {\underline{ 45}}}\\\end{gathered}\end{gathered}$

Note:-

• Prime Factorisation is in Attachment .

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Answer 2

Answer:

Correct answer⬆⬆⬆⬆

Step-by-step explanation: