# What is the least common multiple of 4t to the power of 3 and 6t to the power of 4 ?

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Jan 20, 2018

#### Answer:

Please see the step process below;

#### Explanation:

Given;

${\left(4 t\right)}^{3} = {4}^{3} {t}^{3} = \left(4 \times 4 \times 4\right) \left(t \times t \times t\right) = 64 {t}^{3}$

${\left(6 t\right)}^{4} = {6}^{4} {t}^{4} = \left(6 \times 6 \times 6 \times 6\right) \left(t \times t \times t \times t\right) = 1296 {t}^{4}$

Hence looking for the LCM of both, we need to look for the factors of both as well..

Factor of $\Rightarrow 64 {t}^{3} = 2 t \times 2 t \times 2 t \times 2 \times 2 \times 2$

Factor of $\Rightarrow 1296 {t}^{4} = 2 t \times 2 t \times 2 t \times 2 t \times 3 \times 3 \times 3 \times 3$

Now after finding the factors above, we look for their LCM.

Note: If both values have same factors, we choose only one of it..

For example;

Factor of $\Rightarrow {x}^{4} = x \times x \times x \times x$

Factor of $\Rightarrow {x}^{6} = x \times x \times x \times x \times x \times x$

Indentical ones are in color;

Factor of $\Rightarrow {x}^{4} = \textcolor{b l u e}{x \times x \times x \times x}$

Factor of $\Rightarrow {x}^{6} = \textcolor{b l u e}{x \times x \times x \times x} \times x \times x$

Hence the LCM of ${x}^{4} \mathmr{and} {x}^{6} \Rightarrow \textcolor{b l u e}{x \times x \times x \times x} \times x \times x$
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times \times} \downarrow$
[i.e. Taking the identical ones as a single factor, then adding the reminding factors that follows!]

Now back to our question!

We have the factors;

Factor of $\Rightarrow 64 {t}^{3} = 2 t \times 2 t \times 2 t \times 2 \times 2 \times 2$

Factor of $\Rightarrow 1296 {t}^{4} = 2 t \times 2 t \times 2 t \times 2 t \times 3 \times 3 \times 3 \times 3$

Spotting out the identical factors;

Factor of $\Rightarrow 64 {t}^{3} = \textcolor{b l u e}{2 t \times 2 t \times 2 t} \times 2 \times 2 \times 2$

Factor of $\Rightarrow 1296 {t}^{4} = \textcolor{b l u e}{2 t \times 2 t \times 2 t} \times 2 t \times 3 \times 3 \times 3 \times 3$

Therefore the LCM of $64 {t}^{3} \mathmr{and} 1296 {t}^{4} \Rightarrow \textcolor{b l u e}{2 t \times 2 t \times 2 t} \times 2 t \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$

$\textcolor{w h i t e}{\times \times \times \times \times x} \Rightarrow 16 {t}^{4} \times 8 \times 81$

$\textcolor{w h i t e}{\times \times \times \times \times x} \Rightarrow 10 , 368 {t}^{4}$

Hope this helps!

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Ujjwal Share
Jan 20, 2018

See below

#### Explanation:

$4 {t}^{3} = 2 \cdot 2 \cdot t \cdot t \cdot t$
$6 {t}^{4} = 2 \cdot 3 \cdot t \cdot t \cdot t \cdot t$

Thus,LCM = $2 \cdot 2 \cdot 3 \cdot t \cdot t \cdot t \cdot t$
= $12 {t}^{4}$

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