Lutasin ang frac{24x(x^{3}+1)^{3/2}-3x^{2}(72x^{2}(x^3+1)^7/2)}{16(x^3+1)^3/2}

$$\frac { 24 x ( x ^ { 3 } + 1 ) ^ { 3 / 2 } - 3 x ^ { 2 } ( 72 x ^ { 2 } ( x ^ { 3 } + 1 ) ^ { 7 / 2 } ) } { 16 ( x ^ { 3 } + 1 ) ^ { 3 / 2 } }$$

I-evaluate

$$-\frac{27x^{10}}{2}-27x^{7}-\frac{27x^{4}}{2}+\frac{3x}{2}$$

Maiikling Hakbang sa Solution

Para mag-multiply ng mga power na may parehong base, i-add ang mga exponent ng mga ito. I-add ang 2 at 2 para makuha ang 4.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 72\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

I-multiply ang 3 at 72 para makuha ang 216.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

I-factor ang mga expression na hindi pa na-factor.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

I-cancel out ang 8\left(x^{3}+1\right)^{\frac{3}{2}} sa parehong numerator at denominator.

$$\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2}$$

Palawakin ang expression.

$$\frac{-27x^{10}-54x^{7}-27x^{4}+3x}{2}$$

I-factor

$$\frac{3x\left(1-9x^{3}-18x^{6}-9x^{9}\right)}{2}$$

Mga Hakbang ng Solution

Para mag-multiply ng mga power na may parehong base, i-add ang mga exponent ng mga ito. I-add ang 2 at 2 para makuha ang 4.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 72\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

I-multiply ang 3 at 72 para makuha ang 216.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

I-factor ang mga expression na hindi pa na-factor sa \frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

I-cancel out ang 8\left(x^{3}+1\right)^{\frac{3}{2}} sa parehong numerator at denominator.

$$factor(\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2})$$

Gamitin ang distributive property para i-multiply ang 3x gamit ang -9x^{9}-18x^{6}-9x^{3}+1.

$$factor(\frac{-27x^{10}-54x^{7}-27x^{4}+3x}{2})$$

Isaalang-alang ang -27x^{10}-54x^{7}-27x^{4}+3x. I-factor out ang 3.

$$3\left(-9x^{10}-18x^{7}-9x^{4}+x\right)$$

Isaalang-alang ang -9x^{10}-18x^{7}-9x^{4}+x. I-factor out ang x.

$$x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)$$

I-rewrite ang kumpletong naka-factor na expression. Pasimplehin. Ang polynomial -9x^{9}-18x^{6}-9x^{3}+1 ay hindi naka-factor dahil wala itong anumang rational root.

$$\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2}$$