```Question 217291

The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.

So let's list the factors of 25344

Factors:

1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 18, 22, 24, 32, 33, 36, 44, 48, 64, 66, 72, 88, 96, 99, 128, 132, 144, 176, 192, 198, 256, 264, 288, 352, 384, 396, 528, 576, 704, 768, 792, 1056, 1152, 1408, 1584, 2112, 2304, 2816, 3168, 4224, 6336, 8448, 12672, 25344

Notice how 2304 is the largest perfect square, so lets factor 25344 into 2304*11

{{{sqrt(2304*11)}}} Factor 25344 into 2304*11

{{{sqrt(2304)*sqrt(11)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}

{{{48*sqrt(11)}}} Take the square root of the perfect square 2304 to get 48

So the expression {{{sqrt(25344)}}} simplifies to {{{48*sqrt(11)}}}

In other words, {{{sqrt(25344)=48*sqrt(11)}}}

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Check:

Notice if we evaluate the square root of 25344 with a calculator we get

{{{sqrt(25344)=159.197989937059}}}

and if we evaluate {{{48*sqrt(11)}}} we get

{{{48*sqrt(11)=159.197989937059}}}

This shows that {{{sqrt(25344)=48*sqrt(11)}}}. So this verifies our answer ```