A mathematics teacher bought a birthday cake for his son with the dedication

Happy Birthday!!!
(2 raise to 5/2 .(the dot means times) 2 raise to 3/4 divided by 2 raise to 1/4)th

How old is his son?

kindly help me...

N = 2^(5/2) * 2^(3/4)/2^(1/4).

N = 2^(13/4)/2^(1/4) = 2^(12/4)=2^3=8.

To determine the age of the mathematics teacher's son, you have to simplify the expression mentioned in the dedication of the cake.

The expression is: 2^(5/2) * (2^(3/4)) / 2^(1/4)

Step 1: Simplify the exponents
To simplify the expression, we'll work with the exponents separately.

2^(5/2) can be rewritten as the square root of 2 raised to the power of 5:
2^(5/2) = √(2^5)

2^(3/4) can be rewritten as the fourth root of 2 raised to the power of 3:
2^(3/4) = ⁴√(2^3)

2^(1/4) can also be rewritten as the fourth root of 2:
2^(1/4) = ⁴√2

Step 2: Combine the terms
Now that we have simplified the exponents, we can combine the terms of the expression:

(√(2^5) * ⁴√(2^3)) / ⁴√2

Step 3: Continue simplifying
To simplify further, we can combine the roots:

(√(2^5) * ⁴√(2^3)) / ⁴√2 = √(2^5) * (⁴√(2^3) / ⁴√2)

Step 4: Final simplification
Next, we can simplify each part separately:

√(2^5) = 2^(5/2) = 2^2 * 2^(1/2) = 4 * √2

⁴√(2^3) = 2^(3/4) = 2^(1/4) * 2^(1/4) * 2^(1/4) = √2 * √2 * √2 = 2√2

⁴√2 = √2 (since the fourth root of 2 is the same as the square root of 2)

Now we can rewrite the entire expression:

4 * √2 * 2√2 / √2

Simplifying further, we see that the √2 terms cancel out:

4 * 2 = 8

Therefore, the age of the mathematics teacher's son is 8 years old.

To determine the age of the mathematics teacher's son, we need to understand the notation and simplify the expression in the dedication on the cake.

Let's break down the notation step by step:

1. "2 raise to 5/2" means 2 raised to the power of 5/2. This can be calculated as the square root of 2 raised to the power of 5: √(2^5) = √32.

2. The dot symbol (·) represents multiplication, so "(2 raise to 5/2) · (2 raise to 3/4)" means multiplying the previous result (√32) by 2 raised to the power of 3/4: √32 · (2^(3/4)).

3. Finally, the expression is divided by "2 raise to 1/4." This means dividing the previous result (√32 · (2^(3/4))) by 2 raised to the power of 1/4: (√32 · (2^(3/4))) / (2^(1/4)).

Now, let's simplify the expression step by step:

1. Simplifying √32, we get √(16 × 2) = √16 × √2 = 4√2.

2. Simplifying (2^(3/4)), we can rewrite it as the fourth root of 2 cubed: 2^(3/4) = (2^(1/4))^3 = (2^(1/4))^2 × (2^(1/4)) = 2^(2/4) × 2^(1/4) = √2 × 2^(1/4) = √2 × (√√2).

3. Simplifying (2^(1/4)), we take the fourth root of 2: 2^(1/4) = √(√2).

Now, let's substitute these values back into the expression:

(4√2 · √2 × (√√2)) / (√(√2))

Simplifying further:

(4√2 × 2 × (√√2)) / (√2)

Canceling out the √2 terms:

8 × (√√2)

Therefore, we can conclude that the dedication on the cake simplifies to "Happy Birthday!! 8√√2th."

However, without additional information, we cannot determine the exact age of the mathematics teacher's son based on just this dedication.