Problem 70 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{8}{4 a^{3}} $$

Step-by-Step Solution

Answer

The expression is \( 2 \cdot a^{-3} \).

Step 1: Identify the Expression

The expression given is \( \frac{8}{4a^3} \). This expression is a division problem where 8 is divided by \(4a^3\).

Step 2: Simplify the Denominator

The denominator is \(4a^3\). Since 8 and 4 are both numbers, they can be divided. \(\frac{8}{4} = 2\). So the expression simplifies to \(\frac{2}{a^3}\).

Step 3: Convert Division to Multiplication

To write the expression without a denominator, use the property \( \frac{1}{a^n} = a^{-n} \). So, \( \frac{2}{a^3} = 2 \cdot a^{-3} \).

Step 4: Final Expression in Product Form

The expression \( \frac{8}{4a^3} \) can be expressed in product form as \( 2 \cdot a^{-3} \).

Key Concepts

Quotient to Product

Turning a quotient into a product is a nifty mathematical trick. Imagine you have a fraction like \( \frac{8}{4a^3} \). This is a quotient because it involves division. Quotients can look complicated, but there's a way to simplify them by changing them into products, which use multiplication instead.

When you see \( \frac{1}{a^n} \), you can rewrite it by using its power as a negative exponent. It becomes \( a^{-n} \). So, if you want to change \( \frac{8}{4a^3} \) into a product, you'd first remove the fraction by dividing \( 8 \) by \( 4 \) to get \( 2 \). Then, you replace the division by \( a^3 \) with multiplication by using a negative exponent, so \( a^3 \) turns into \( a^{-3} \).

The original problem is now rewritten without a denominator: \( 2 \cdot a^{-3} \). Changing quotients to products will solve a lot of algebra problems and make your calculations easier!

Negative Exponents

Negative exponents can seem confusing, but they have a really straightforward meaning. If you have \( a^{-n} \), it means the reciprocal of \( a^n \). In simple terms, \( a^{-n} = \frac{1}{a^n} \). Negative signs just tell us to flip the exponent into a fraction.

When converting a quotient into a product, negative exponents play a crucial role. For example, in our expression \( 2 \cdot a^{-3} \), the \( a^{-3} \) arises from using the rule that \( \frac{1}{a^3} = a^{-3} \). This switch helps transform division into multiplication. It's like magic, but it's backed up by algebraic rules!

This technique simplifies calculations and helps you avoid fractions, making the mathematics much neater and easier to handle in the long run.

Simplifying Expressions

Simplifying expressions means making them as simple as possible. It’s like decluttering a room: removing what you don’t need and organizing what's left so everything looks neat and tidy. In algebra, simplifying can involve reducing fractions, collecting like terms, or converting complex divisions into easier multiplications.

Let’s look at \( \frac{8}{4a^3} \) again. By simplifying, first we divide numbers: \( \frac{8}{4} = 2 \). With no denominator left, we handle \( a^3 \) by using negative exponents: \( \frac{1}{a^3} = a^{-3} \).

Now, our simple expression is \( 2 \cdot a^{-3} \). This is much easier to read, write, and calculate with other expressions. The goal is to make life simpler, and understanding how to simplify expressions is a major tool in any math student’s toolbox!