Problem 72 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{56 a^{\frac{1}{6}}}{8 a^{\frac{1}{4}}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{a^{\frac{1}{12}}}\).

Step 1: Simplify the Fraction

Start by dividing the numerical coefficients. Here, divide 56 by 8. \[\frac{56}{8} = 7\]The fraction simplifies to:\[7 \cdot \frac{a^{\frac{1}{6}}}{a^{\frac{1}{4}}}\]

Step 2: Apply the Quotient Rule to Exponents

Apply the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\). Subtract the exponents of \(a\):\[7 \cdot a^{\frac{1}{6} - \frac{1}{4}}\]

Step 3: Simplify the Exponent

To subtract the exponents, find a common denominator. The least common denominator for 6 and 4 is 12, so:\[a^{\frac{1}{6} - \frac{1}{4}} = a^{\frac{2}{12} - \frac{3}{12}}\]Now perform the subtraction:\[a^{\frac{2}{12} - \frac{3}{12}} = a^{-\frac{1}{12}}\]

Step 4: Express with Positive Exponents

Convert the negative exponent into a positive exponent by taking the reciprocal:\[7 \cdot a^{\frac{-1}{12}} = \frac{7}{a^{\frac{1}{12}}}\]

Key Concepts

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number or variable. For example, an exponentiation such as \(a^3\) is simply \(a\) multiplied by itself three times: \(a \times a \times a\). When dealing with fractions and roots, exponents can become fractions themselves, like \(a^{\frac{1}{2}}\) which represents the square root of \(a\). Working with exponents often involves rules such as the product, power, and quotient rules which simplify calculations. Understanding these foundational concepts is crucial for solving algebraic expressions efficiently.

Quotient Rule

The quotient rule for exponents is a critical tool for simplifying fractions that involve variables with exponents. This rule states that when you divide like bases, you subtract the exponents:

• \(\frac{a^m}{a^n} = a^{m-n}\)

This is useful for expressions where both the numerator and the denominator have the same base. For instance, if you have \(\frac{a^{\frac{1}{6}}}{a^{\frac{1}{4}}}\), you simply subtract the exponent in the denominator from the exponent in the numerator. This simplifies the expression and makes further calculations more straightforward. Always remember to apply the quotient rule to each variable separately if multiple variables are involved.

Positive Exponents

In algebra, it is common practice to express terms using positive exponents, as they are generally simpler to interpret and use. Negative exponents indicate an inverse or reciprocal. For instance, \(a^{-n}\) is the same as \(\frac{1}{a^n}\). To convert a negative exponent to a positive one, you can take the reciprocal of the base and make the exponent positive. In the context of our example, converting \(7 \cdot a^{-\frac{1}{12}}\) to a positive exponent results in \(\frac{7}{a^{\frac{1}{12}}}\). This results in a more straightforward and easier to comprehend expression.

Fraction Simplification

Simplifying fractions is about reducing expressions to their most basic form. This often includes dividing the numerators and denominators by their greatest common factor (GCF). In mathematical expressions involving variables, it's also about using algebraic rules to combine and reduce terms.

• First, reduce the numerical coefficients. In our problem, 56 divided by 8 simplifies to 7.
• Then, use the quotient rule to simplify variables with exponents.

Once you've simplified both the coefficients and variables, the fraction is in its simplest form. This two-step process of numerical and exponential simplification results in clearer and more manageable algebraic expressions.