Given:

The given number is √180.

Calculation:

According to the question, we have,

√180 = √(3 × 3 × 2 × 5 × 2)

⇒ √180 = 3 × 2√5

⇒ 6√5

∴ The value of √180 is 6√5.

Calculation:

$\frac{\sqrt[3]{x}}{2.56}=\frac{100}{x}$

$x^{4/3}=100\times 2.56$

$x={256}^{3/4}$

x = 16^{2 × 3/4} = 4^{2 × 3/2}

x = 4³ = 64

Additional InformationFormula used:

am × an = a(m + n)

am ÷ an = a(m - n)

(am)n = amn

Given:

16√n = √1600 + √576

Calculation:

According to the question,

⇒ 16√n = √1600 + √576

⇒ 16√n = 40 + 24

⇒ √n = 64/16

⇒ √n = 4

n = (4)² = 16

∴ Value of n is 16

Formula used:

(a^x)^y = a^xy

Calculation:

${\left[5{(8^{\frac{1}{3}}+{27}^{\frac{1}{3}})}^3\right]}^{\frac{1}{4}}$

⇒ ${\left[5{((2^3{)}^{\frac{1}{3}}+(3^3{)}^{\frac{1}{3}})}^3\right]}^{\frac{1}{4}}$

⇒ ${\left[5{((2)+(3))}^3\right]}^{\frac{1}{4}}$

⇒ ${\left[5{(2+3)}^3\right]}^{\frac{1}{4}}$ 

⇒ ${\left[5\times {\left(5\right)}^3\right]}^{\frac{1}{4}}$

⇒ ${\left[{\left(5\right)}^4\right]}^{\frac{1}{4}}$ = 5

∴ The value of ${\left[5{(8^{\frac{1}{3}}+{27}^{\frac{1}{3}})}^3\right]}^{\frac{1}{4}}$ is 5.

Concept used:

a^b × a^c = a^{b + c}

If a^b = a^c then b = c

(a^b)^c = a^{b × c}

Calculation:

1252 × (625)-0.25 = 5x

⇒ 5^{3 × 2} × 5^{4 × -0.25} = 5^x

⇒ 5⁶ × 5^-1 = 5^x

⇒ 5^{6 - 1} = 5^x

⇒ 5⁵ = 5x

⇒ x = 5

Given,

2³ × 3⁴ × 1080 ÷ 15 = 6^x

⇒ 2³ × 3⁴ × 72 = 6^x

⇒ 2³ × 3⁴ × (2 × 6²) = 6^x

⇒ 2⁴ × 3⁴ × 6² = 6^x

⇒ (2 × 3)⁴ × 6² = 6^x           [∵ x^m × y^m = (xy)^m]

⇒ 6⁴ × 6² = 6^x

⇒ 6^{(4 + 2)} = 6^x

⇒ x = 6

(56/x) = √(196/324)​

⇒ (56/x) = 14/18

⇒ x = (56 × 18)/14

⇒ x = 72

∴ The value of x is 72.

∛3375

∛(3 × 3 × 3 × 5 × 5 × 5)

3 × 5 = 15

Given:

The square of 2112 

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

(2100 + 12)²

⇒ (2100)² + (12)² + 2 × 2100 × 12

⇒ 4410000 + 144 + 50400

⇒ 4460544

∴ The square of 2112 is 4460544.

Concept used:

Calculations:

⇒ 292 - 252 = 63 + x

⇒ 841 - 625 = 63 + x

⇒ 216 = 63 + x

⇒ 216 - 216 = x

⇒ x = 0

∴ The value of x is 0.