Question Simplify the ratio $\sqrt{32} : \sqrt{50}$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the ratio of two square roots: $\sqrt{32}$ and $\sqrt{50}$ .
2. We need to simplify this ratio to its simplest form.

STEP 2

First, we will simplify each square root separately by factoring them into prime factors.
$\sqrt{32} = \sqrt{2^5}$ $\sqrt{50} = \sqrt{2 \times 5^2}$

STEP 3

Now, we will use the property of square roots that allows us to take the square root of a product as the product of the square roots.
$\sqrt{32} = \sqrt{2^5} = \sqrt{2^4} \times \sqrt{2} = 2^2 \times \sqrt{2} = 4\sqrt{2}$ $\sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2} = \sqrt{2} \times 5 = 5\sqrt{2}$

STEP 4

Now that we have simplified both square roots, we can write the original ratio with these simplified forms.
$\sqrt{32}: \sqrt{50} = 4\sqrt{2} : 5\sqrt{2}$

STEP 5

We notice that both terms in the ratio have a common factor of $\sqrt{2}$ . We can divide both terms by $\sqrt{2}$ to simplify the ratio further.
$4\sqrt{2} : 5\sqrt{2} = \frac{4\sqrt{2}}{\sqrt{2}} : \frac{5\sqrt{2}}{\sqrt{2}}$

STEP 6

Simplify the ratio by canceling out the common factor of $\sqrt{2}$ .
$\frac{4\sqrt{2}}{\sqrt{2}} : \frac{5\sqrt{2}}{\sqrt{2}} = 4 : 5$

STEP 7

The ratio is now in its simplest form.
$\sqrt{32}: \sqrt{50} = 4 : 5$
The simplified form of the ratio is $4:5$ .

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