What is the square root of 16^36
The Correct Answer and Explanation is:
To find the square root of 163616^{36}1636, we can use the property of exponents and square roots.
Step 1: Express 163616^{36}1636 in terms of powers of 2
First, note that 161616 is a power of 2:16=2416 = 2^416=24
So we can rewrite 163616^{36}1636 as:1636=(24)3616^{36} = (2^4)^{36}1636=(24)36
Step 2: Apply the power of a power rule
The power of a power rule states that (am)n=am×n(a^m)^n = a^{m \times n}(am)n=am×n. Applying this rule:(24)36=24×36=2144(2^4)^{36} = 2^{4 \times 36} = 2^{144}(24)36=24×36=2144
Thus, 1636=214416^{36} = 2^{144}1636=2144.
Step 3: Find the square root
The square root of a number is equivalent to raising it to the power of 12\frac{1}{2}21. So to find the square root of 21442^{144}2144, we write:2144=2144×12=272\sqrt{2^{144}} = 2^{144 \times \frac{1}{2}} = 2^{72}2144=2144×21=272
Final Answer
The square root of 163616^{36}1636 is 2722^{72}272.
Explanation
By first converting 16 into its base-2 equivalent, 16=2416 = 2^416=24, we can express 163616^{36}1636 as 21442^{144}2144. Then, taking the square root of 21442^{144}2144 involves using the rule of exponents that simplifies it to 2722^{72}272. This approach ensures that we can easily handle large exponents in exponential expressions, avoiding unnecessary complications.
