Question

`\frac{(3 √(5)) {}^(3) + 3 \sqrt{ {5}^(3) } }{ √(30 * 24) - \sqrt[4]{25 * 16} }`

simplify the following expression​

Answer 1

The simplified expression is
`\frac{150 √(5) }{4√(15)-4\sqrt[4]{10} }` .

Simplify the numerator:

`(3√(5) )^3`: Use the power of a power rule:
`(a^m)^n = a^{(m* n)`

This gives us

`(3^3)(√(5))^3 \\= 27 * 5^((3/2)) \\= 27 * 5√(5) }`

`3(√( 5^3))` : Use the constant multiple rule:
`k * a = k * (√((a)))`

This gives us
`3 * 5^((3/2)) = 3 * 5^{(1.5)`

Combine the terms in the numerator:

`27 * 5√(5) + 3 * 5√(5) \\= 135 √(5) + 15√(5) \\= 150√(5)`

Simplify the denominator:

- Factor the radicals in both terms:

`-√(30 * 24)\\=√(30) \cdot √(24)\\=√(2 \cdot 3 \cdot 5) \cdot √(2 \cdot 2 \cdot 2 \cdot 3)\\=2 √(3 \cdot 5) \cdot 2 √(2 \cdot 3)\\=4 √(15) \\`

`& -\sqrt[4]{25 * 16}\\=\sqrt[4]{25} \cdot \sqrt[4]{16}\\=\sqrt[4]{5 \cdot 5} \cdot \sqrt[4]{2 \cdot 2 \cdot 2 \cdot 2}\\=2 \sqrt[4]{5} \cdot 2 \sqrt[4]{2}\\=4 \sqrt[4]{10}`

Substitute the simplified numerator and denominator into the fraction:

`\frac{150 √(5) }{4√(15)-4\sqrt[4]{10} }`