Problem 30 - Algebra 2

Question

Evaluate each expression. $$ 8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}} $$

Step-by-Step Solution

Answer

The expression evaluates to 4096.

Step 1: Apply the Product of Powers Rule

When multiplying like bases, add the exponents. For the expression \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}}\), add the exponents \(\frac{3}{2}\) and \(\frac{5}{2}\): \[ \frac{3}{2} + \frac{5}{2} = \frac{3+5}{2} = \frac{8}{2} \] This simplifies to 4.

Step 2: Simplify the Expression

Now that we've added the exponents, replace the initial expression with the simplified version that uses the sum of the exponents:\[ 8^{\frac{8}{2}} = 8^4 \]

Step 3: Calculate the Power of 8

Calculate \(8^4\), which involves multiplying 8 by itself 4 times:\[ 8^4 = 8 \times 8 \times 8 \times 8 \]Start by calculating step-by-step: 1. \(8 \times 8 = 64\) 2. \(64 \times 8 = 512\) 3. \(512 \times 8 = 4096\).

Step 4: Conclusion

Thus, the value of the expression \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}}\) is 4096.

Key Concepts

Product of Powers Rule

When it comes to exponentiation, one useful shortcut is the "Product of Powers Rule." This rule is especially handy when dealing with expressions that involve the multiplication of like bases. In these cases, you do not need to compute each power separately. Instead, this rule allows you to simply add the exponents together.
For example, let's consider the expression \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}}\). The base \(8\) is common to both terms, which means we can apply the Product of Powers Rule.
By adding the exponents \(\frac{3}{2}\) and \(\frac{5}{2}\), we get:

• \(\frac{3}{2} + \frac{5}{2} = \frac{8}{2}\)

Simplifying \(\frac{8}{2}\) gives \(4\), which means \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}} = 8^4\). This helps reduce complex calculations significantly, especially with larger numbers.

Simplifying Expressions

When you simplify expressions in mathematics, you make them easier to work with without changing their values. In our example, after applying the Product of Powers Rule, we simplified the expression \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}}\) to \(8^4\).
Simplification involves several strategies, such as combining like terms, which we've done by adding the exponents.

• By converting \(8^{\frac{3}{2}} \cdot 8^{\frac{5}{2}}\) into a simpler form \(8^4\), the problem becomes much more manageable.

Understanding how to simplify expressions accurately is crucial because it sets the stage for solving or evaluating more complex math problems. It removes unnecessary complications, allowing you to focus on core computations.

Calculating Powers

After simplifying an expression like \(8^4\), the next step is calculating this power. Calculating powers means multiplying the base number (8 in this case) by itself a certain number of times corresponding to the exponent.
For \(8^4\), you multiply 8 by itself 4 times:

• First, calculate \(8 \times 8 = 64\).
• Next, multiply the result by 8: \(64 \times 8 = 512\).
• Finally, multiply by 8 again: \(512 \times 8 = 4096\).

These steps illustrate not only the process but also the importance of organizing your calculations.
Breaking it down in distinct phases helps ensure accuracy, especially with large numbers. Once you complete this calculation, you arrive at the final answer: \(4096\).