Question

simplify by dividing each term in the numerator by the denominator.\\(\frac{15r^{10} - 5r^{8} + 40r^{2}}{5r^{4}}\\)\\(\bigcirc\\ 3r^{6} - r^{4} + 8r^{2}\\)\\(\bigcirc\\ 3r^{6} - r^{4} + \frac{8}{r^{2}}\\)\\(\bigcirc\\ 3r^{14} - r^{12} + \frac{8}{r^{6}}\\)\\(\bigcirc\\ \frac{1}{3r^{6} - r^{4} + 8r^{2}}\\)

Explanation:

Step1: Divide each term in numerator by denominator

We have the expression \(\frac{15r^{10}-5r^{8}+40r^{2}}{5r^{4}}\). We can split this into three separate fractions: \(\frac{15r^{10}}{5r^{4}}-\frac{5r^{8}}{5r^{4}}+\frac{40r^{2}}{5r^{4}}\)

Step2: Simplify each fraction

• For the first fraction \(\frac{15r^{10}}{5r^{4}}\), divide the coefficients \(15\div5 = 3\) and use the rule of exponents \(a^{m}\div a^{n}=a^{m - n}\) for \(r\): \(r^{10-4}=r^{6}\), so the first term is \(3r^{6}\)
• For the second fraction \(\frac{5r^{8}}{5r^{4}}\), divide the coefficients \(5\div5=1\) and use the exponent rule: \(r^{8 - 4}=r^{4}\), so the second term is \(-r^{4}\) (the negative sign is carried over)
• For the third fraction \(\frac{40r^{2}}{5r^{4}}\), divide the coefficients \(40\div5 = 8\) and use the exponent rule: \(r^{2-4}=r^{- 2}=\frac{1}{r^{2}}\), so the third term is \(\frac{8}{r^{2}}\) (since \(r^{-n}=\frac{1}{r^{n}}\))

Combining these simplified terms, we get \(3r^{6}-r^{4}+\frac{8}{r^{2}}\)

Answer:

\(3r^{6}-r^{4}+\frac{8}{r^{2}}\) (corresponding to the option \(3r^{6}-r^{4}+\frac{8}{r^{2}}\))