Question

which expression is equivalent to \\(\left(4x^{5}+11\
ight)^{2}\\)?\
\\(\bigcirc\\) \\(16x^{5}+121\\)\
\\(\bigcirc\\) \\(16x^{10}+121\\)\
\\(\bigcirc\\) \\(16x^{10}+88x^{5}+121\\)\
\\(\bigcirc\\) \\(16x^{25}+88x^{5}+121\\)

Explanation:

Step1: Recall the formula for squaring a binomial

The formula for \((a + b)^2\) is \(a^2 + 2ab + b^2\). Here, \(a = 4x^5\) and \(b = 11\).

Step2: Calculate \(a^2\)

For \(a = 4x^5\), \(a^2=(4x^5)^2 = 4^2\times(x^5)^2=16x^{10}\) (using the power of a product rule \((ab)^n=a^n b^n\) and power of a power rule \((a^m)^n=a^{mn}\)).

Step3: Calculate \(2ab\)

Substitute \(a = 4x^5\) and \(b = 11\) into \(2ab\). So, \(2ab = 2\times(4x^5)\times11=88x^5\).

Step4: Calculate \(b^2\)

For \(b = 11\), \(b^2 = 11^2=121\).

Step5: Combine the terms

Putting it all together using the binomial square formula: \((4x^5 + 11)^2=a^2+2ab + b^2=16x^{10}+88x^5 + 121\).

Answer:

\(16x^{10}+88x^5 + 121\) (the third option)