Problem 38 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \left(4^{2} \cdot 5^{2}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Distribute the Exponent

Use the property \( (a \cdot b)^{n} = a^{n} \cdot b^{n} \) of exponents to distribute the square root to both terms inside the parentheses. Thus, the expression becomes \( (4^{2})^{1/2} \cdot (5^{2})^{1/2} \)

Step 2: Simplify the Exponents

Simplify the exponents by multiplying them, as per the rule \( (a^{n})^{m} = a^{n \cdot m} \). The expression becomes \( 4^{2 \cdot 1/2} \cdot 5^{2 \cdot 1/2} \), which further simplifies to \( 4^{1} \cdot 5^{1} \).

Step 3: Evaluate the Expression

Now, solve the expression \( 4^{1} \cdot 5^{1} = 4 \cdot 5 \) to obtain the final solution.

Step 4: Compute the Multiplication

The multiplication \( 4 \cdot 5 \) equals 20.

Key Concepts

Distribute Exponents

When you encounter an expression like \( (a \cdot b)^n \), the exponent can be applied to each factor inside the parentheses separately. This is called distributing the exponent.

• The property used is \( (a \cdot b)^n = a^n \cdot b^n \).

For our original expression \( (4^2 \cdot 5^2)^{1/2} \), we apply the exponent \( 1/2 \) to both \( 4^2 \) and \( 5^2 \).
This transforms the expression into \( (4^2)^{1/2} \cdot (5^2)^{1/2} \).
Understanding this step is crucial as it sets the stage for simplifying each component of the expression individually.

Simplify Exponents

Once you've distributed the exponents, the next step is to simplify them. This uses the rule \( (a^n)^m = a^{n \cdot m} \).

• This allows you to multiply the exponents together.
• For example, \( (4^2)^{1/2} \) becomes \( 4^{2 \cdot 1/2} \).
• Similarly, \( (5^2)^{1/2} \) becomes \( 5^{2 \cdot 1/2} \).

These both simplify the expression to \( 4^1 \cdot 5^1 \), as multiplying 2 by \( 1/2 \) results in 1.
This simplification is vital in making the problem more manageable and closer to evaluation.

Multiplication of Powers

After simplifying the exponents, you're left with the task of multiplying these powers. The outcome of simplifying your exponents in our exercise resulted in \( 4^1 \cdot 5^1 \).

• This is equivalent to simply \( 4 \cdot 5 \), since any number to the power of 1 is the number itself.

Thus, the multiplication becomes a straightforward calculation.
Always remember that multiplying powers that have been reduced in this way simplifies the entire process. By dealing with lowered exponents, it becomes easier to handle the computation involved.

Evaluate Expressions

This final step is where you complete the calculation and evaluate the expression entirely.
For the expression \( 4^1 \cdot 5^1 \), it simplifies directly to \( 4 \cdot 5 \).

• Multiplying these two numbers yields \( 20 \).

Evaluation requires applying basic multiplication now that the powers have been dealt with.
This is usually the most straightforward step when the previous simplifications are correct.
The ability to evaluate an expression correctly completes your understanding of managing exponentiation in the context of distributed and simplified powers.