Question

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{36}{44}$$

Short answer

\(\frac{36}{44} = \frac{9}{11}\). The fraction in lowest terms without negative exponents is \(\frac{9}{11}\).

Step-by-step solution

Prime Factorization

Find the prime factors of the numerator and the denominator. For 36, the prime factors are 2, 2, 3, 3, which can be written as \(2^2 \times 3^2\). For 44, the prime factors are 2, 2, 11, which can be written as \(2^2 \times 11\).

Simplifying the Fraction

Cancel out the common terms in the numerator and the denominator. Both 36 and 44 have a common prime factor of \(2^2\). Cancel these out to simplify the fraction: \(\frac{3^2}{11}\).

Express in Lowest Terms

Write the fraction with the cancelled out terms. The lowest term of the given fraction is \(\frac{9}{11}\), which is already in its simplest form and does not contain any negative exponents.

Key concepts

Prime Factorization

Imagine you're at a party, and your trick to impress everyone is breaking down numbers into their building blocks, you'd be doing what mathematicians call prime factorization. This nifty technique involves expressing a number as a product of prime numbers, which are the numbers that can only be divided by 1 and themselves without leaving a remainder.

Using our party trick on the number 36 from our exercise, we would break it down into 2 x 18, then further into 2 x 2 x 9, until we can't go any further with whole numbers, ending up with 2 x 2 x 3 x 3, or in fancy math terms, as \(2^2 \times 3^2\). This is like uncovering the basic DNA of the number 36, which is crucial for the magic trick we'll perform in the next steps: simplifying algebraic fractions.

Reducing Fractions to Lowest Terms

Now that we know our numbers inside out thanks to prime factorization, we get to play the matchmaker by pairing up common 'genes' from our fraction's family tree, which is really just finding the common factors in the numerator and denominator.

In the example of our fraction \(\frac{36}{44}\), once we've exposed its prime factors, we see a matching pair of \(2^2\) on the top and bottom—like finding out your date is into the same quirky '80s band as you. It's a perfect match, so we can cancel those common factors out. What's left is the fraction in its simplest form, \(\frac{9}{11}\), similar to two singles at a party, now standing out on their own but having a much clearer relationship.

Eliminating Negative Exponents

Negative exponents in math can be perplexing, like receiving a text message full of cryptic shorthand. But fear not! Our goal here is to make sure we express our final answer without any of these confusing characters.

In our example, we were lucky—our simplified fraction \(\frac{9}{11}\) doesn't include any negative exponents. But if we encountered one, the trick would be to reciprocate (flip over) the part of the fraction with the negative exponent. For instance, if our fraction was \(x^{-1}/y^{-2}\), we'd turn it into \(y^2/x\) to make it positive, which is math's way of making everything nice and friendly again. No stress, no mess—just a fraction that's easy to understand and work with.