Problem 590 - Elementary Algebra

Question

In the following exercises, simplify. $$ 4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The simplified form is 64.

Step 1: Apply the Product of Powers Rule

The product of powers rule states that when multiplying two exponents with the same base, you add the exponents. The base here is 4. So, we have:\[4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} = 4^{\left(\frac{5}{2} + \frac{1}{2}\right)}\]

Step 2: Add the Exponents

Add the exponents \(\frac{5}{2}\) and \(\frac{1}{2}\):\[\frac{5}{2} + \frac{1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3\]

Step 3: Simplify the Expression

Now substitute the result of the addition back into the exponent, giving:\[4^3\]

Step 4: Compute the Final Value

Calculate \(4^3\):\[4^3 = 4 \cdot 4 \cdot 4 = 64\]

Key Concepts

Product of Powers Rule

The product of powers rule is a fundamental concept in exponentiation. It states that when you multiply two exponents with the same base, you simply add their exponents. This works because you are essentially combining the repeated multiplications.
For example:
\(a^m \cdot a^n = a^{m+n}\)
In the exercise, we applied this rule to the base 4:
\[4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} = 4^{(\frac{5}{2} + \frac{1}{2})}\]
This addition of exponents reduces the problem to just dealing with one exponent, simplifying the calculation process significantly.

Simplifying Exponents

Simplifying exponents is a necessary skill to manage complex expressions easily. In this case, after applying the product of powers rule, we simplified the expression further by adding the exponents.
Adding exponents like \(\frac{5}{2}\) and \(\frac{1}{2}\) involves using basic fraction addition rules. We can express these fractions with a common denominator and then add them together:
\[\frac{5}{2} + \frac{1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3\]
Thus, the exponent becomes 3, transforming \(4^{\frac{6}{2}}\) into \(4^3\). Removing fractions from the exponent makes further calculations straightforward.

Adding Fractions

Adding fractions is a basic yet crucial arithmetic skill. When adding fractions, you add the numerators (the top numbers) while keeping the same denominator (the bottom number).
For instance, in our exercise, we added \(\frac{5}{2}\) and \(\frac{1}{2}\):
\[\frac{5}{2} + \frac{1}{2} = \frac{5+1}{2} = \frac{6}{2}\]
Once we have the sum, simplifying the resulting fraction to its simplest form gives us the next step for our exponentiation. In this case, \(\frac{6}{2}\) simplifies to 3, making the expression much easier to compute.