Question

express in simplest radical form given ( x > 0 ).( 6x^{2}sqrt{80x}+xsqrt{45x^{3}} )

Explanation:

Step1: Simplify each radical separately

First, simplify \( \sqrt{80x} \). We can factor 80 into \( 16\times5 \), so:
\( \sqrt{80x}=\sqrt{16\times5x}=\sqrt{16}\times\sqrt{5x}=4\sqrt{5x} \)
Then, simplify \( \sqrt{45x^3} \). Factor 45 into \( 9\times5 \) and \( x^3 \) into \( x^2\times x \), so:
\( \sqrt{45x^3}=\sqrt{9\times5\times x^2\times x}=\sqrt{9}\times\sqrt{x^2}\times\sqrt{5x}=3x\sqrt{5x} \) (since \( x>0 \), \( \sqrt{x^2}=x \))

Step2: Substitute the simplified radicals back into the original expression

For the first term \( 6x^2\sqrt{80x} \), substitute \( \sqrt{80x}=4\sqrt{5x} \):
\( 6x^2\times4\sqrt{5x}=24x^2\sqrt{5x} \)
For the second term \( x\sqrt{45x^3} \), substitute \( \sqrt{45x^3}=3x\sqrt{5x} \):
\( x\times3x\sqrt{5x}=3x^2\sqrt{5x} \)

Step3: Combine like terms

Now we have the expression \( 24x^2\sqrt{5x}+3x^2\sqrt{5x} \). Since both terms have \( x^2\sqrt{5x} \), we can combine the coefficients:
\( (24 + 3)x^2\sqrt{5x}=27x^2\sqrt{5x} \)

Answer:

\( 27x^2\sqrt{5x} \)