Simplifying the Expression: 3(2a)^3/2 ^2
When it comes to simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS) and apply the rules of exponents. In this article, we'll tackle the expression 3(2a)^3/2 ^2 and break it down step by step to arrive at its simplified form.
Step 1: Evaluate the Exponentiation
The expression 3(2a)^3/2 ^2 can be rewritten as:
3(2a)^(3/2) ^2
To evaluate the exponentiation, we need to apply the rule of exponents that states a^(m/n) = (a^m)^(1/n). Therefore, we can rewrite the expression as:
3((2a)^3)^(1/2) ^2
Step 2: Simplify the Inner Expression
Now, let's simplify the inner expression (2a)^3. Using the rule of exponents that states (ab)^m = a^m b^m, we can expand the expression as:
(2a)^3 = 2^3 a^3
= 8a^3
So, the expression becomes:
3(8a^3)^(1/2) ^2
Step 3: Evaluate the Square Root
Next, we need to evaluate the square root of 8a^3. Using the rule of exponents that states a^(1/2) = √a, we can rewrite the expression as:
3√(8a^3) ^2
= 3√(2^3 a^3) ^2
= 3(2a)^(3/2) ^2
Step 4: Simplify the Outer Exponentiation
Finally, we need to simplify the outer exponentiation (2a)^(3/2) ^2. Using the rule of exponents that states (a^m)^n = a^(mn), we can simplify the expression as:
(2a)^(3/2) ^2 = (2a)^(3/2 × 2)
= (2a)^3
So, the final simplified expression is:
3(2a)^3
In conclusion, by following the order of operations and applying the rules of exponents, we were able to simplify the expression 3(2a)^3/2 ^2 to its final form, 3(2a)^3.
