Rule:  a^b = (√a)^(2*b)

The rule can be used when solving equations.

like for example:

Solve for x:

x^x  =  64^4

Rule:  64^4  =  8^8  ➔

x^x  =  8^8  ➔

x  =  8

The rule also works in the opposite direction, of course.

a^b  =  (a²)^(b/2)

Note that this rule can also be stacked:

65536^2  =  256^4  =  16^8  =  4^16  =  2^32

The rule is also the predecessor of this rule:

Rule:

Positive numbers a, b

The (a)th root of b    ...equals the (2•a)th root of b²

b^(1/a)  =  (b²)^(1/(2a))

Thank you for sharing these interesting properties of exponents and roots. Let's break down the rules you presented and see how they can be applied in various scenarios:

### Rule: \( a^b = (\sqrt{a})^{2b} \)

This rule essentially states that raising a number to a power can be rewritten by taking the square root of the base and doubling the exponent. Let's see some applications and examples:

#### Example 1: 
Solve for \( x \):
\[ x^x = 64^4 \]

Using the rule:
\[ 64^4 = (8^2)^4 = 8^8 \]

Thus:
\[ x^x = 8^8 \]

So:
\[ x = 8 \]

#### Example 2:
The rule also works in the opposite direction:
\[ a^b = (a^2)^{b/2} \]

For instance:
\[ 256^4 = (16^2)^4 = 16^8 \]

#### Example 3:
Stacking the rule:
\[ 65536^2 = (256^2)^2 = 256^4 \]
\[ 256^4 = (16^2)^4 = 16^8 \]
\[ 16^8 = (4^2)^8 = 4^{16} \]
\[ 4^{16} = (2^2)^{16} = 2^{32} \]

### Rule: 
Positive numbers \( a, b \)
\[ b^{1/a} = (b^2)^{1/(2a)} \]

This rule states that the \(a\)th root of \(b\) is equal to the \((2 \cdot a)\)th root of \(b^2\). Let's look at an example:

#### Example 4:
Find the 4th root of 16:
\[ 16^{1/4} = (16^2)^{1/(2 \cdot 4)} = 256^{1/8} \]

To verify, compute the values:
\[ 16^{1/4} = 2 \]
\[ 256^{1/8} = 2 \]

Both values are indeed equal, verifying the rule.

Would you like to see more examples or explanations of these rules? Here are five questions you might consider asking next:

1. How can these rules be applied to logarithms?
2. Can you show more complex examples involving these rules?
3. How do these rules relate to the properties of logarithms?
4. What are some practical applications of these exponent and root rules?
5. How do these rules apply to solving exponential equations in different contexts?

**Tip:** Remember that understanding the manipulation of exponents and roots can greatly simplify solving complex algebraic and calculus problems. Keep practicing different scenarios to master these concepts.

Explore the rules of exponents and roots, including practical examples and applications. Learn how to solve equations using rules such as a^b = (sqrt(a))^(2b) and b^(1/a) = (b^2)^(1/(2a)). Ideal for students and enthusiasts interested in advanced algebraic concepts.

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