Question

Simplify.
Remove all perfect squares from inside the square root.
✓112a^6

Answer 1

`4√(7)a^3`

Explanation:

`√(112a^6) \\\\\mathrm{Apply\:radical\:rule\:}\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}\\\\=√(112)√(a^6)\\\\=4√(7)√(a^6)\\\\\mathrm{Apply\:exponent\:rule}:\quad \:a^(bc)=\left(a^b\right)^c\\\\a^6=a^(3*\:2)=\left(a^3\right)^2\\\\=4√(7)√(\left(a^3\right)^2)\\\\\mathrm{Apply\:radical\:rule\:}\sqrt[n]{a^n}=a\\\\√(\left(a^3\right)^2)=a^3\\\\=4√(7)a^3`

Answer 2

The
`\( √(112a^6) \)` simplifies to
`\( 4a^3 √(7) \)`.

To simplify
`\( √(112a^6) \)`, you can break down the expression into its prime factorization and then identify the perfect squares.

`\[ √(112a^6) = √(2^4 \cdot 7 \cdot a^6) \]`

Now, separate the perfect squares and non-perfect squares:

`\[ √(2^4 \cdot 7 \cdot a^6) = √((2^2)^2 \cdot 7 \cdot (a^3)^2) \]`

Now, take the square root of the perfect squares:

`\[ 2^2 \cdot a^3 \cdot √(7) \]`

Combine the terms:

`\[ 4a^3 √(7) \]`

So,
`\( √(112a^6) \)` simplifies to
`\( 4a^3 √(7) \)`.