Problem 94 - Algebra and Trigonometry

Question

Simplify using properties of exponents. $$ \frac{72 x^{\frac{3}{4}}}{9 x^{\frac{1}{3}}} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{72 x^{\frac{3}{4}}}{9 x^{\frac{1}{3}}}\) is \(8x^{\frac{5}{12}}\).

Step 1: Simplify Coefficients

Start by simplifying the coefficients 72 and 9 by division. This will give \(\frac{72}{9} = 8\).

Step 2: Apply Property of Exponents to Variables

When you divide terms with the same base, you subtract the exponent of the divisor from the exponent of the dividend. Thus, \(x^{\frac{3}{4}}\) divided by \(x^{\frac{1}{3}}\) results in \(x^{\frac{3}{4}-\frac{1}{3}}\).

Step 3: Simplify the Exponential Expression

It's now necessary to subtract the fractions. Convert the fractions to have the same denominator, which is 12 in this case. This results in \(x^{\frac{9}{12}-\frac{4}{12}} = x^{\frac{5}{12}}\).

Key Concepts

Simplifying Fractions

Simplification of fractions is like cleaning things up. It aims to make fractions easier to understand or use. When simplifying a fraction like \(\frac{72}{9}\), you're basically asking how many times the bottom number (the denominator) fits into the top number (the numerator). Start by dividing both numbers by their greatest common divisor, which is 9 in this case.

• Divide 72 by 9: This gives you 8.
• Hence, \(\frac{72}{9} = 8\).

By simplifying in this way, you bring the fraction down to its simplest form, which helps to make the entire expression less complex.

Division of Exponents

Dividing numbers with exponents might sound tricky, but it's quite simple once you know the rule: subtract the exponents when the bases are the same. For instance, let's look at \(x^{\frac{3}{4}}\) and \(x^{\frac{1}{3}}\). Both have the base \(x\), so you can apply the rule of subtracting the exponents.
To divide, follow these steps:

• Take \(\frac{3}{4}\) (from the numerator exponent).
• Subtract \(\frac{1}{3}\) (from the denominator exponent).

Before you subtract, ensure both fractions have the same denominator. This might involve finding a common denominator, but don't worry; we'll cover that next. In our equation:

• The operation changes to: \(\frac{3}{4} - \frac{1}{3}\).
• With a little fraction conversion, this difference becomes \(x^{\frac{5}{12}}\).

This is how division of terms with exponents simplifies the expression.

Common Denominators

Fractions are a little easier to manage when they have the same bottom number or denominator. When we handle exponents like fractions, a common denominator is key. Let's say you need to subtract \(\frac{3}{4}\) from \(\frac{1}{3}\). The first step is to find a number both 4 and 3 can agree on. This is called the least common denominator.

• Here, both denominators can agree on 12.
• Convert \(\frac{3}{4}\) to an equivalent fraction: It's \(\frac{9}{12}\).
• Convert \(\frac{1}{3}\) to the same denominator: It becomes \(\frac{4}{12}\).

Now, you can subtract them:

• \(\frac{9}{12} - \frac{4}{12} = \frac{5}{12}\).

Finding a common denominator makes calculating easier and helps bring two fractions into a friendly format for subtraction. Understanding this will make fraction operations, especially in exponents, simpler to navigate.