Problem 88 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\frac{72 x^{3 / 4}}{9 x^{1 / 3}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Coefficients

Start with simplifying the numerical coefficient. Divide \(72\) by \(9\) to get \(8\). The fraction simplifies to \(8 x^{3/4} / x^{1/3}\)

Step 2: Apply the Division Property of Exponents

Apply the division rule of exponents. It states that when you divide two powers with the same base, you subtract the exponents. In this case, one would obtain \(8x^{3/4 - 1/3}\)

Step 3: Subtract the Exponents

Subtract the exponents. Make sure both fractions have the same denominator (which is \(12\) in this case), then subtract the numerators. Subtracting the exponents \(9/12 - 4/12\) gives \(8x^{5/12}\)

Step 4: Simplify the Answer

Finally, \(5/12\) cannot be simplified further. So the final answer stays as \(8x^{5/12}\)

Key Concepts

Division Property of Exponents

When working with exponents, one important property is the division property. This property simplifies expressions where exponents with the same base are divided by subtracting the exponent of the denominator from the exponent of the numerator.
For instance, given a fraction like \[\frac{x^a}{x^b}\], you simplify it to \[x^{a-b}\]. The key point is to ensure the bases are identical before applying this property.
In our example, notice that both the numerator and the denominator have the same base, 'x'. Therefore, we apply the division property: \[\frac{x^{3/4}}{x^{1/3}} = x^{3/4 - 1/3}\]. This subtraction results in the new exponent for base 'x', which will be further simplified. Using this property can greatly reduce the complexity of expressions involving exponents.

Simplifying Exponents

Simplifying exponents often involves basic arithmetic operations such as addition, subtraction, and even finding a common denominator when working with fraction exponents.
Instead of handling complicated numerators and denominators, you can streamline the expression by reducing these components.
In our exercise, when \[3/4 - 1/3\]was calculated, both terms needed a common denominator to be subtracted easily. By setting the denominator to 12, the exponents are rewritten as \[9/12\]and \[4/12\]. The subtraction results in \[9/12 - 4/12 = 5/12\]. Simplifying the exponents is crucial in achieving the cleanest and most manageable form of expressions, which is particularly useful when variables are present.

Fraction Exponents

Fraction exponents represent roots and powers simultaneously and can initially seem complex, but they follow set rules that make them manageable. A fractional exponent, such as \[x^{3/4}\], indicates that the base 'x' is both taken to a power and rooted. Specifically, the numerator is the power the base is raised to, and the denominator represents the root. This means \[x^{3/4}\] is equivalent to taking the fourth root of \[x^3\].
They are particularly useful when simplifying expressions that require root extraction or power calculations.
Managing fraction exponents involves understanding both the operations for powers and roots while performing basic arithmetic to simplify wherever feasible. In our solved example, after applying the division property of exponents, the fractional exponent becomes \[x^{5/12}\]. This outcome combines the fractional operations of both powers and roots into one simplified exponent, representing a streamlined version of the original rational expression.