Problem 68 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{15 w^{2}}{9 w^{5}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{15 w^{2}}{9 w^{5}}\) is \(\frac{5}{3w^3}\).

Step 1: Simplify fraction

First of all, reduce the fraction \(\frac{15}{9}\) to its simplest form: \(\frac{5}{3}\). Thus, the given expression becomes \(\frac{5 w^{2}}{3 w^{5}}\).

Step 2: Simplify the power terms using the quotient of powers property

For the same base \(w\) we get \(w^{2-5} = w^{-3}\). Thus, the simplified expression now becomes \(\frac{5}{3} * w^{-3}\).

Step 3: Convert the negative exponent to positive

Because negative exponents are not permitted, convert \(w^{-3}\) to \(\frac{1}{w^3}\). Thus, the final simplified expression will be \(\frac{5}{3w^3}\).

Key Concepts

Simplifying Fractions

When you come across a fraction in algebra, simplifying it makes the rest of the problem easier to solve. To simplify a fraction, you divide the numerator (the top number) and the denominator (the bottom number) by the greatest common factor they share. For example, if you have \(\frac{15}{9}\), both 15 and 9 can be divided by 3, which is their greatest common factor. After dividing both by 3, you are left with \(\frac{5}{3}\), a much simpler fraction.

Simplifying fractions is like tidying up a room before you start studying; it clears out the unnecessary and leaves you with just what you need to focus on. In the context of our exercise, simplifying the fraction was the first step which paved the way for easier manipulation of the algebraic expression that followed.

Quotient of Powers Property

Imagine you have a tower of blocks, and each block represents a power of a number. If you have two such towers and you need to compare them, you can subtract the height of the second tower from the first to find out how much taller the first one is. This is similar to what we do with the quotient of powers property in algebra.

Essentially, when you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Using our problem as an example, we take the exponents for \(w\) which are 2 for the numerator and 5 for the denominator, and subtract them (\(2-5\)) to get \(-3\).

This results in \(w^{-3}\), which illustrates that \(w^2\) is much smaller compared to \(w^5\). The quotient of powers property is a shortcut that keeps us from having to write out and divide long strings of multiplication. It significantly simplifies the process of working with expressions containing powers.

Negative Exponents

In algebra, negative exponents can seem intimidating at first, but they follow a simple rule: a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. To put it another way, anything with a negative exponent can be flipped to the denominator of a fraction and made positive. For example, \(w^{-3}\) is the same as \(\frac{1}{w^3}\).

We apply this rule in the final step of our exercise to turn \(w^{-3}\) into a form that's easier to work with. It helps us avoid dealing with negative exponents and puts the expression in a more familiar and interpretable form. Consequently, the once negative exponent is now helping us see how the variable \(w\) is being divided into the expression, manifesting as \(\frac{5}{3w^3}\) in its most simplified form.