Question

rewrite the expression using only positive exponents, and simplify. (assume that any variables in the expression are nonzero.) \\( x^8(4x^0y^6)(8y)^0 \\)

evaluate the expression without using a calculator. (enter your answer in scientific notation.) \\( (9 \times 10^4)^2 \\) \\( \times 10 \\)

Explanation:

Step1: Apply power of a product rule

We know that \((ab)^n = a^n b^n\). So for \((9\times10^{4})^{2}\), we can apply this rule:
\((9\times10^{4})^{2}=9^{2}\times(10^{4})^{2}\)

Step2: Calculate each part

First, calculate \(9^{2}\), we get \(9^{2} = 81\).
Then, for \((10^{4})^{2}\), we use the power of a power rule \((a^{m})^{n}=a^{mn}\), so \((10^{4})^{2}=10^{4\times2}=10^{8}\).
Now we have \(81\times10^{8}\), but we need to write it in scientific notation. In scientific notation, the coefficient should be between \(1\) and \(10\). So we rewrite \(81\) as \(8.1\times10^{1}\).
Then we have \(8.1\times10^{1}\times10^{8}\). Using the rule \(a^{m}\times a^{n}=a^{m + n}\), we get \(8.1\times10^{1 + 8}=8.1\times10^{9}\)

Answer:

\(8.1\times10^{9}\)