Problem 57 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \frac{24 x^{3} y^{5}}{32 x^{7} y^{-9}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac {3x^{4}y^{14}}{4} \)

Step 1: Simplify the coefficients

The numerical coefficients 24 and 32 can be reduced to their simplest form. Both can be divided by 8, leading to \(3/4\).

Step 2: Simplify the base x

Applying the law of exponents for division \(a^{n} / a^{m} = a^{n - m}\) for base \(x\), the exponents are subtracted \(3 - 7 = -4\). So, \(x^{3}/x^{7}=x^{-4}\). Considering the negative exponent rule which converts \(x^{-4}\) to \(1/x^{4}\), but this is in the denominator of the original expression so it becomes \(x^{4}\) in the numerator.

Step 3: Simplify the base y

Repeat the step above for base \(y\). So, \(y^5/y^{-9}\) becomes \(y^{5 - (-9)} = y^{14}\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing a given expression to its simplest form. This process often includes minimizing complex fractions, eliminating parentheses, or consolidating terms that are alike.
In algebraic contexts, simplifying can involve several steps, such as:

• Reducing coefficients, which are the numerical parts of terms.
• Applying laws of operations to variables with exponents.
• Simplifying fractions by cancelling out common factors.

This makes mathematical expressions easier to understand and work with, allowing for quicker calculations in further steps.

Negative Exponents

Understanding negative exponents is crucial to simplifying expressions. A negative exponent means that the base is on the wrong side of a fraction line. To correct this, you flip it. For instance, if you have \(a^{-n}\), you change it to \(\frac{1}{a^n}\).
This rule is especially useful when simplifying fractions where variables have negative exponents. By moving the variable from the numerator to the denominator (or vice versa), the exponent becomes positive.

• Example: If \(x^{-3}\) is in the numerator, you can move it to the denominator as \(\frac{1}{x^3}\).
• Remember: Negative exponents do not make numbers negative!

This principle helps in managing expressions more efficiently.

Law of Exponents

The law of exponents provides fundamental guidelines for manipulating exponential expressions. These rules allow us to multiply, divide, and raise powers of expressions with exponents. The key laws to remember include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: \((a^m)^n = a^{m \times n}\).

Using these laws effectively simplifies the process of dealing with complex exponential expressions. For example, when dividing like bases, simply subtract the exponents, which can help reduce the complexity of the expression.
Applying these rules systematically can transform a cumbersome expression into a user-friendly form.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and the denominator contain algebraic expressions. Simplifying these fractions follows similar steps as rational numbers.

The process usually involves:

• Simplifying coefficients by finding common factors.
• Applying the laws of exponents to variables.
• Canceling out common factors in both the numerator and the denominator.

Through these steps, you can make algebraic fractions much easier to work with. In our original exercise, simplifying the expression involved reducing coefficients from 24 and 32 to 3/4 and managing the exponents of the variables to clear the fraction.
You achieve a clearer, more succinct expression that is far easier to evaluate or use in further calculations.