Problem 68 - Algebra 2

Question

Simplify each expression. \(8^{\frac{1}{2}} \cdot 40^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

The simplification of the expression \(8^{\frac{1}{2}} \cdot 40^{\frac{1}{2}}\) is approximately 17.89.

Step 1: Understanding the Exponent of 1/2

The exponent of 1/2 represents the square root of a number. Therefore, \(8^{\frac{1}{2}}\) equates to \(\sqrt{8}\) and \(40^{\frac{1}{2}}\) equates to \(\sqrt{40}\).

Step 2: Calculation of the Square Root

Calculate the square root of 8 and 40. \(\sqrt{8}=2.83\) (approx) and \(\sqrt{40}=6.32\) (approx). These results are rounded to the nearest hundredths.

Step 3: Multiplication of the Results

Multiplying the results of both square roots (2.83 and 6.32), we obtain approximately 17.88.

Step 4: Simplify the Original Expression

Instead of dealing with approximations, let's simplify the original expression accurately. As the \(8^{\frac{1}{2}}\) and \(40^{\frac{1}{2}}\) both have an exponent of 1/2, we can \(8*40\) first, and applying square root at last. This would keep our values accurate. So, we have \((8*40)^{\frac{1}{2}} = 320^{\frac{1}{2}}\). This simplifies to 17.89 (rounded to nearest hundredths).

Key Concepts

Understanding Square Roots

Square roots can sometimes be confusing, but they are an essential part of simplifying expressions. The square root of a number is the value that, when multiplied by itself, gives the original number. In the given exercise, the expressions like \(8^{\frac{1}{2}}\) and \(40^{\frac{1}{2}}\) are forms of square roots.
This is because the exponent \(\frac{1}{2}\) is another way to indicate taking the square root.For example:

• \(8^{\frac{1}{2}} = \sqrt{8}\)
• \(40^{\frac{1}{2}} = \sqrt{40}\)

Instead of finding the approximated decimal values, these expressions can also be understood and manipulated in their square root form, which keeps them exact until a final calculation is needed.

Expression Simplification Techniques

Expression simplification often involves breaking down more complex expressions into simpler parts. This allows for easier manipulation and evaluation. In the given problem, after realizing that both components \(8^{\frac{1}{2}}\) and \(40^{\frac{1}{2}}\) can be written as square roots, we can further simplify their multiplication.First, you can combine the expressions under a single square root, like this:

• Combine the product of the numbers: \( (8 \cdot 40)^{\frac{1}{2}} \)
• Simplify under the square root: \(320^{\frac{1}{2}}\)

This approach makes the calculation cleaner and reduces the need to handle intermediate decimal approximations, helping to prevent rounding errors.
It's an effective way to retain accuracy until finalizing the expression simplification.

Mastering Exponent Rules

Understanding exponent rules is crucial for simplifying expressions effectively. Exponents allow us to express large numbers in a concise way and manipulate expressions effortlessly. There are key rules to remember when dealing with exponents, especially fractional ones, like in our exercise.Some rules include:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Fractional exponents represent roots: \(a^{1/n} = \sqrt[n]{a}\)

In the given problem, applying the rule for fractional exponents lets us understand that \(a^{\frac{1}{2}}\) is equivalent to \(\sqrt{a}\). Knowing these rules helps in interpreting expressions in different forms, making it easier to carry out computation or further simplification steps accurately.