Question

which expression is equivalent to \\(\left(4h^{7}k^{2}\
ight)^{4}\\)?
\\(\bigcirc\\) \\(16h^{11}k^{6}\\)
\\(\bigcirc\\) \\(16h^{28}k^{8}\\)
\\(\bigcirc\\) \\(256h^{11}k^{6}\\)
\\(\bigcirc\\) \\(256h^{28}k^{8}\\)

Explanation:

Step1: Apply the power - of - a - product rule

The power - of - a - product rule states that \((ab)^n=a^n\times b^n\). For the expression \((4h^{7}k^{2})^{4}\), we can apply this rule as follows:
\((4h^{7}k^{2})^{4}=4^{4}\times(h^{7})^{4}\times(k^{2})^{4}\)

Step2: Calculate the power of the constant and the powers of the variables

• Calculate \(4^{4}\): We know that \(4\times4 = 16\), \(16\times4=64\), \(64\times4 = 256\), so \(4^{4}=256\).
• Calculate \((h^{7})^{4}\): According to the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\), when \(a = h\), \(m = 7\) and \(n = 4\), we have \((h^{7})^{4}=h^{7\times4}=h^{28}\).
• Calculate \((k^{2})^{4}\): Using the power - of - a - power rule again, with \(a = k\), \(m = 2\) and \(n = 4\), we get \((k^{2})^{4}=k^{2\times4}=k^{8}\).

Step3: Combine the results

Multiply the results from Step 2 together: \(4^{4}\times(h^{7})^{4}\times(k^{2})^{4}=256\times h^{28}\times k^{8}=256h^{28}k^{8}\)

Answer:

256\(h^{28}\)\(k^{8}\) (the fourth option: 256\(h^{28}\)\(k^{8}\))