Question

rewrite .1 as 10 times a power of 10. .1 = 10 · 10□

Explanation:

Step1: Recall exponent rules

We know that \(a^m\times a^n=a^{m + n}\) and \(0.1=\frac{1}{10}=10^{- 1}\), and \(10 = 10^1\). Let the unknown power be \(x\), so \(10\times10^x=10^{1 + x}\).

Step2: Set up equation

We want \(10\times10^x = 0.1=10^{-1}\). Then, by the rule \(a^m\times a^n=a^{m + n}\), we have \(10^{1 + x}=10^{-1}\). Since the bases are the same (\(a = 10\)), the exponents must be equal. So \(1+x=-1\).

Step3: Solve for x

Subtract 1 from both sides of the equation \(1+x=-1\). We get \(x=-1 - 1=-2\).

Answer:

\(-2\)