Solve (-9k^2/4k^2+3)^2-4-6k/4k^2+3

$$( \frac { - 9 k ^ { 2 } } { 4 k ^ { 2 } + 3 } ) ^ { 2 } - 4 \frac { - 6 k } { 4 k ^ { 2 } + 3 }$$

Evaluate

$$\frac{3k\left(27k^{3}+32k^{2}+24\right)}{\left(4k^{2}+3\right)^{2}}$$

Short Solution Steps

To raise \frac{-9k^{2}}{4k^{2}+3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-4\times \frac{-6k}{4k^{2}+3}$$

Express 4\times \frac{-6k}{4k^{2}+3} as a single fraction.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{4\left(-6\right)k}{4k^{2}+3}$$

Multiply 4 and -6 to get -24.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4k^{2}+3\right)^{2} and 4k^{2}+3 is \left(4k^{2}+3\right)^{2}. Multiply \frac{-24k}{4k^{2}+3} times \frac{4k^{2}+3}{4k^{2}+3}.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}}$$

Since \frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}} and \frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-9k^{2}\right)^{2}-\left(-24k\left(4k^{2}+3\right)\right)}{\left(4k^{2}+3\right)^{2}}$$

Expand \left(-9k^{2}\right)^{2}.

$$\frac{\left(-9\right)^{2}\left(k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-9\right)^{2}k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

Calculate -9 to the power of 2 and get 81.

$$\frac{81k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4k^{2}+3\right)^{2} and 4k^{2}+3 is \left(4k^{2}+3\right)^{2}. Multiply \frac{-24k}{4k^{2}+3} times \frac{4k^{2}+3}{4k^{2}+3}.

$$\frac{81k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}}$$

Since \frac{81k^{4}}{\left(4k^{2}+3\right)^{2}} and \frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{81k^{4}-\left(-24k\left(4k^{2}+3\right)\right)}{\left(4k^{2}+3\right)^{2}}$$

Do the multiplications in 81k^{4}-\left(-24k\left(4k^{2}+3\right)\right).

$$\frac{81k^{4}+96k^{3}+72k}{\left(4k^{2}+3\right)^{2}}$$

Expand \left(4k^{2}+3\right)^{2}.

$$\frac{81k^{4}+96k^{3}+72k}{16k^{4}+24k^{2}+9}$$

Expand

$$\frac{3\left(27k^{4}+32k^{3}+24k\right)}{\left(4k^{2}+3\right)^{2}}$$

Short Solution Steps

To raise \frac{-9k^{2}}{4k^{2}+3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-4\times \frac{-6k}{4k^{2}+3}$$

Express 4\times \frac{-6k}{4k^{2}+3} as a single fraction.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{4\left(-6\right)k}{4k^{2}+3}$$

Multiply 4 and -6 to get -24.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4k^{2}+3\right)^{2} and 4k^{2}+3 is \left(4k^{2}+3\right)^{2}. Multiply \frac{-24k}{4k^{2}+3} times \frac{4k^{2}+3}{4k^{2}+3}.

$$\frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}}$$

Since \frac{\left(-9k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}} and \frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-9k^{2}\right)^{2}-\left(-24k\left(4k^{2}+3\right)\right)}{\left(4k^{2}+3\right)^{2}}$$

Expand \left(-9k^{2}\right)^{2}.

$$\frac{\left(-9\right)^{2}\left(k^{2}\right)^{2}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-9\right)^{2}k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

Calculate -9 to the power of 2 and get 81.

$$\frac{81k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k}{4k^{2}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4k^{2}+3\right)^{2} and 4k^{2}+3 is \left(4k^{2}+3\right)^{2}. Multiply \frac{-24k}{4k^{2}+3} times \frac{4k^{2}+3}{4k^{2}+3}.

$$\frac{81k^{4}}{\left(4k^{2}+3\right)^{2}}-\frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}}$$

Since \frac{81k^{4}}{\left(4k^{2}+3\right)^{2}} and \frac{-24k\left(4k^{2}+3\right)}{\left(4k^{2}+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{81k^{4}-\left(-24k\left(4k^{2}+3\right)\right)}{\left(4k^{2}+3\right)^{2}}$$

Do the multiplications in 81k^{4}-\left(-24k\left(4k^{2}+3\right)\right).

$$\frac{81k^{4}+96k^{3}+72k}{\left(4k^{2}+3\right)^{2}}$$

Expand \left(4k^{2}+3\right)^{2}.

$$\frac{81k^{4}+96k^{3}+72k}{16k^{4}+24k^{2}+9}$$