Exponents and Exponential Functions

Below is an equation that is not true.

$${{{{{10}0}^{1\over2}}}=50}$$

a.   Why is the statement incorrect? What do you think the correct value of $${{{{10}0}^{1\over2}}}$$ is?

b.   Consider the following pattern. Where does $${{{{10}0}^{1\over2}}}$$ fit in?

$${{{10}0}^3=1,000,000}$$

$${{{10}0}^2={10},000}$$

$${{{10}0}^1={{10}0}}$$

$${{{10}0}^0=1}$$

c.   Consider rewriting the base $${{10}0}$$ as a power of $${10}$$. How does this shed light on the value of $${{{{10}0}^{1\over2}}}$$?

$${{{{10}0}^{1\over2}}}=(\square)^{1\over2}$$

d.   Try out these other rational exponents:

$${25^{1\over2}}$$                    $${144^{1\over2}}$$                   $${8^{1\over3}}$$

All of the following equations are true.

$${\sqrt{x}=x^{1\over2}}$$                $${\sqrt[3]{x}=x^{1\over3}}$$                $${(\sqrt{x})^2=x}$$               $${x^{2\over3}=\sqrt[3]{x^2}}$$

Determine a general statement to represent the relationship between a radical and its exponential expression.

Write the radicals in exponential form and write the exponentials in radical form. 

a.   $${5^{6\over5}}$$

b.   $${4^{-{2\over3}}}$$

c.   $${2n^{2\over5}}$$

d.   $${\sqrt[3]{7^2}}$$

e.   $${1\over{\sqrt[3]{5}}}$$

f.   $${\sqrt{(3x)^5}}$$

Henry explains why $${4^{3\over2}=8}$$:

"I know that $${4^3}$$ is $${{64}}$$ and the square root of $${{64}}$$ is $$8$$."

Here is Henrietta’s explanation for why $${4^{3\over2}=8}$$:

"I know that $${\sqrt4=2}$$ and the cube of $$2$$ is $$8$$. "

1. Are Henry and Henrietta correct? Explain. 
2. Calculate $$4^{5\over2}$$ and $$27^{2\over3}$$ using Henry’s or Henrietta’s strategy. 
3. Use both Henry and Henrietta’s reasoning to express $${x^{m\over n}}$$ using radicals (here $$m$$ and $$n$$ are positive integers and we assume $${x>0}$$).