Problem 15 - College Algebra

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. $$ 4^{2} \cdot 4^{3} \div 4^{-4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(4^{9}\).

Step 1: Apply the Product of Powers Property

When you have the same base being multiplied with exponents, the exponents are added together. Here, we first apply this to the multiplication part of the expression: \[4^{2} \cdot 4^{3} = 4^{2+3} = 4^{5}\]So, the expression simplifies to \(4^5 \div 4^{-4}\).

Step 2: Apply the Quotient of Powers Property

When you divide terms with the same base, subtract the exponent of the denominator from the exponent of the numerator: \[4^{5} \div 4^{-4} = 4^{5 - (-4)} = 4^{5 + 4}\]This simplifies to \(4^{9}\).

Step 3: Ensure Positive Exponents

The expression \(4^{9}\) already has a positive exponent, so no further steps are needed to adjust the exponents.

Key Concepts

Product of Powers Property

The product of powers property is a crucial concept for working with exponents. When you multiply terms that have the same base, adding their exponents is the key to simplifying such expressions. Imagine you have an expression like \(4^2 \cdot 4^3\). Here, both parts of the expression share the base \(4\). This allows us to simplify by adding the exponents:

• Add the exponents: \(2 + 3 = 5\).
• The expression becomes \(4^5\). This gives you a single base with a single exponent, which is much simpler.

Next time, see if your bases match! If they do, just add the exponents and simplify. This will make working with exponents much easier.

Quotient of Powers Property

When you're dividing exponents with the same base, the rule is similar but involves subtraction. This is known as the quotient of powers property. If you come across something like \( \frac{4^5}{4^{-4}} \), all you need to do is:

• Subtract the exponent in the denominator from the exponent in the numerator: \(5 - (-4) = 5 + 4 = 9\).
• This simplifies the expression to \( 4^9 \), a more straightforward representation.

Remember, the key here is subtraction. Subtraction can sometimes feel a bit tricky, especially with negative numbers involved. But with practice, it will become second nature.

Positive Exponents

Positive exponents are more intuitive to work with than negative ones. A positive exponent simply tells you how many times to multiply a number by itself. In our example, \(4^9\) indicates multiplying \(4\) by itself 9 times.Negative exponents, on the other hand, imply taking the reciprocal and raising it to the positive of that exponent. Working with negative exponents can often lead to confusion. But when you have a positive exponent, there's no need for extra steps to simplify further.If your solution requires positive exponents, always look for ways to rearrange or modify your expression until all exponents are positive. This makes your work neater and easier to understand.