1Cademy - Simplifying $$\frac{3}{4}\sqrt{192}

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Adding and Subtracting Like Square Roots

Example

Simplifying $\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108}$

Simplify the expression $\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108}$ , where both terms have fractional coefficients and the radicals must be simplified.

Simplify each radical by extracting the largest perfect square factor:

$\frac{3}{4}\sqrt{192} = \frac{3}{4} \cdot \sqrt{64} \cdot \sqrt{3} = \frac{3}{4} \cdot 8 \cdot \sqrt{3} = 6\sqrt{3}$

$\frac{5}{6}\sqrt{108} = \frac{5}{6} \cdot \sqrt{36} \cdot \sqrt{3} = \frac{5}{6} \cdot 6 \cdot \sqrt{3} = 5\sqrt{3}$

In each term, the fractional coefficient is multiplied by the integer from the simplified radical, producing a whole-number coefficient.

Rewrite the expression:

$6\sqrt{3} - 5\sqrt{3}$

Both terms contain $\sqrt{3}$ , so combine the coefficients: $6 - 5 = 1$ .

$\frac{3}{4}\sqrt{192} - \frac{5}{6}\sqrt{108} = \sqrt{3}$

This example extends the technique to fractional coefficients. When the integer from the simplified radical multiplies with the fraction, the result may simplify to a whole number — here $\frac{3}{4} \cdot 8 = 6$ and $\frac{5}{6} \cdot 6 = 5$ — making the final combination straightforward.

Updated 2026-04-21

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OpenStax Elementary Algebra

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