Question

please give me answer fast​

Answer 1

$\large \bold{ \sqrt[3]{1728} - \sqrt[3]{216} }$

$= \sqrt[3]{ {12}^{3} } - \sqrt[3]{ {6}^{3} }$

$\bold{ = 12 ^{3 \times \frac{1}{3} } - {6}^{3 \times \frac{1}{3} } }$

$= 12 - 6$

$= \color{blue}\bold{6}$

Answer 2

Answer :-

6

Given to find the value of :-

$\sqrt[3]{1728} - \sqrt[3]{216}$

Formulae to know :-

$\red{ \sqrt[n]{a} = a {}^{ \dfrac{1}{n} }}$

$\red{a {}^{m} \times b {}^{m} \times c {}^{m} = (abc) {}^{m}}$

$\red{(a {}^{m}) {}^{n} = a {}^{m n}}$

Solution:-

Firstly lets write the radicals in simplest form or by prime factorisation method we shall write them in product of prime factors .

$\begin{gathered}\begin{array}{c | c} \underline2 &\underline{1728} \\ \underline2&\underline{864} \\ \underline2 &\underline{432}\\ \underline2&\underline{216} \\ \underline2&\underline{108}\\ \underline2&\underline{54}\\ \underline3&\underline{27} \\ \underline3&\underline{9}\\ \underline3&\underline3 \\ &1\end{array}\end{gathered}$

So,

1728 = 2 × 2× 2 × 2 × 2 ×2 ×3 ×3 ×3

1728 = 2⁶ × 3³

1728 = 2³ × 2³ × 3³

$\begin{gathered}\begin{array}{c | c} \underline2&\underline{216} \\ \underline2&\underline{108}\\ \underline2&\underline{54}\\ \underline3&\underline{27} \\ \underline3&\underline{9}\\ \underline3&\underline3 \\ &1\end{array}\end{gathered}$

216 = 2 ×2 × 2 × 3×3 ×3

216 = 2³ × 3³

i.e writing the powers with respect to 3 in order to cancel the radical

$\sqrt[3]{1728} - \sqrt[3]{216}$

$= \sqrt[3]{2 {}^{3} \times 2 {}^{3} \times 3 {}^{3} } - \sqrt[3]{2 {}^{3} \times 3 {}^{3} }$

By using this formula

$\red{a {}^{m} \times b {}^{m} \times c {}^{m} = (abc) {}^{m}}$

$= \sqrt[3]{(2 \times 2 \times 3) {}^{3} } - \sqrt[3]{(2 \times 3) {}^{3} }$

$= \sqrt[3]{(12) {}^{3} } - \sqrt[3]{(6) {}^{3} }$

By using this formula

$\red{ \sqrt[n]{a} = a {}^{ \dfrac{1}{n} }}$

$= (12 {}^{3} ) {}^{ \frac{1}{3} } - (6 {}^{3}) {}^{ \frac{1}{3} }$

By using this formula

$\red{(a {}^{m}) {}^{n} = a {}^{m n}}$

$= 12 {}^{3 \times \frac{1}{3} } - 6 {}^{3 \times \frac{1}{3} }$

$= 12 {}^{ \not3 \times \frac{1}{ \not3} } - 6 {}^{ \not3 \times \frac{1}{ \not3} }$

$= 12 - 6$

$\red{= 6}$

So,

$\red{\sqrt[3]{1728} - \sqrt[3]{216}=6}$