Question

find the smallest number by 13720 must be multiplied to obtain a perfect cube

Answer 1

The smallest number by 13720 must be multiplied to obtain a perfect cube is 25.

Given:

• A number
• 13720

To find: Find the smallest number by 13720 must be multiplied to obtain a perfect cube.

Solution:

We know that,

If a number have prime factors with powers multiples of 3, then the number be a perfect cube.

i.e.

$\sqrt[3]{ {a}^{3} } = a$

In order to find the smallest number to multiply, follow these simple steps.

Step 1:

Find prime factors of 13720.

$13720 = 2 \times 2 \times 2 \times 5 \times 7 \times 7 \times 7$

or

$13720 = {2}^{3} \times 5 \times {7}^{3}$

Step 2:

Multiply the number by 25.

As two of the prime factors of 13720 already have power 3.

But 5 didn't.

So,

$13720 \times 25 = {2}^{3} \times 5 \times 25 \times {7}^{3}$

or

$13720 \times 25 = {2}^{3} \times 5 \times {5}^{2} \times {7}^{3}$

or

$13720 \times 25 = {2}^{3} \times {5}^{3} \times {7}^{3}$

as,

${a}^{n} \times {b}^{n} = ( {ab)}^{n}$

So,

$13720 \times 25 = {(2 \times 5 \times 7)}^{3}$

or

$13720 \times 25 = ({70})^{3}$

Thus,

25 is the smallest number by which 13720 must be multiplied to obtain a perfect cube.

i.e. 13720×25 = 343,000 is a perfect cube.

Learn more:

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(i)3

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(iii)5

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Answer 2

Step-by-step explanation:

The smallest number by 13720 must be multiplied to obtain a perfect cube is 25.

Given:

A number13720

To find: Find the smallest number by 13720 must be multiplied to obtain a perfect cube.

Solution:

We know that,

If a number have prime factors with powers multiples of 3, then the number be a perfect cube.

i.e.

\begin{gathered} \sqrt[3]{ {a}^{3} } = a \\ \end{gathered}3a3=a

In order to find the smallest number to multiply, follow these simple steps.

Step 1:

Find prime factors of 13720.

\begin{gathered}13720 = 2 \times 2 \times 2 \times 5 \times 7 \times 7 \times 7 \\ \end{gathered}13720=2×2×2×5×7×7×7

or

\begin{gathered}13720 = {2}^{3} \times 5 \times {7}^{3} \\ \end{gathered}13720=23×5×73

Step 2:

Multiply the number by 25.

As two of the prime factors of 13720 already have power 3.

But 5 didn't.

So,

\begin{gathered}13720 \times 25 = {2}^{3} \times 5 \times 25 \times {7}^{3} \\ \end{gathered}13720×25=23×5×25×73

or

\begin{gathered}13720 \times 25 = {2}^{3} \times 5 \times {5}^{2} \times {7}^{3} \\ \end{gathered}13720×25=23×5×52×73

or

\begin{gathered}13720 \times 25 = {2}^{3} \times {5}^{3} \times {7}^{3} \\ \end{gathered}13720×25=23×53×73

as,

\begin{gathered} {a}^{n} \times {b}^{n} = ( {ab)}^{n} \\ \end{gathered}an×bn=(ab)n

So,

\begin{gathered}13720 \times 25 = {(2 \times 5 \times 7)}^{3} \\ \end{gathered}13720×25=(2×5×7)3

or

\begin{gathered}13720 \times 25 = ({70})^{3} \\ \end{gathered}13720×25=(70)3

Thus,

25 is the smallest number by which 13720 must be multiplied to obtain a perfect cube.

i.e. 13720×25 = 343,000 is a perfect cube.

Learn more:

1) The smallest number by which 1944 should be multiplied so that it becomes a

perfect cube is-

(i)3

(11) 2

(iii)5

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2) by what least number 3750 must be multiplied to give a perfect square. Fast tell me it's important

https://brainly.in/question/48611817