What is factor 9x^{-3/2}+6x^{-1/2}+x^{1/2} ?

The solution to factor 9x^{-3/2}+6x^{-1/2}+x^{1/2} is ((x+3)^2)/(x^{3/2)}

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factor 9x^{-3/2}+6x^{-1/2}+x^{1/2}

factor

9 x^{- \frac{3}{2}} + 6 x^{- \frac{1}{2}} + x^{\frac{1}{2}}

\frac{( x + 3 ) ^{2}}{x ^{\frac{3}{2}}}

Solution steps

9 x^{- \frac{3}{2}} + 6 x^{- \frac{1}{2}} + x^{\frac{1}{2}}

Rewrite as

= 3 \cdot 3 x^{- \frac{3}{2}} + 3 \cdot 2 x^{- \frac{1}{2}} + x^{\frac{1}{2}}

Factor out common term

3

= 3 (3 x^{- \frac{3}{2}} + 2 x^{- \frac{1}{2}}) + x^{\frac{1}{2}}

Factor

3 x^{- \frac{3}{2}} + 2 x^{- \frac{1}{2}} : \frac{3 + 2 x}{x ^{\frac{3}{2}}}

= x^{\frac{1}{2}} + 3 \cdot \frac{2 x + 3}{x ^{\frac{3}{2}}}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

x^{- \frac{1}{2}} = x^{1 +- \frac{3}{2}} = x^{1} x^{- \frac{3}{2}}, x^{\frac{1}{2}} = x^{2 +- \frac{3}{2}} = x^{2} x^{- \frac{3}{2}}

= 9 x^{- \frac{3}{2}} + 6 x^{1} x^{- \frac{3}{2}} + x^{2} x^{- \frac{3}{2}}

Factor out common term

x^{- \frac{3}{2}}

= x^{- \frac{3}{2}} (9 + 6 x + x^{2})

Factor

x^{2} + 6 x + 9 : (x + 3)^{2}

= x^{- \frac{3}{2}} (x + 3)^{2}

Refine

= \frac{( x + 3 ) ^{2}}{x ^{\frac{3}{2}}}

- 6 x + 24

9 - ∣ 2 x + 7 ∣ \leq 6

- \frac{1}{2} + 3 \leq x + 1 \leq \frac{1}{2} + 3

(- 3 \frac{1}{2})^{2}

\frac{1 ^{2} + 4 + 1}{( 1 ^{2} + 1 + 1 ) ^{2}}

What is factor 9x^{-3/2}+6x^{-1/2}+x^{1/2} ?
The solution to factor 9x^{-3/2}+6x^{-1/2}+x^{1/2} is ((x+3)^2)/(x^{3/2)}