Problem 62 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(\frac{-49 u^{3} v^{4}}{-7 u^{4} v^{7}}\right)^{-1} $$

Step-by-Step Solution

Answer

\( \frac{u v^3}{7} \).

Step 1: Simplify the Fraction

Start by simplifying the fraction inside the parentheses. The original expression is \( \frac{-49 u^3 v^4}{-7 u^4 v^7} \). First, the negative signs cancel out: \( \frac{49 u^3 v^4}{7 u^4 v^7} \). Then, divide the coefficients: \( \frac{49}{7} = 7 \). The simplified expression is \( 7 \frac{u^3 v^4}{u^4 v^7} \).

Step 2: Simplify the Variables

Now simplify the variables by applying the rule \( \frac{a^m}{a^n} = a^{m-n} \): \( \frac{u^3}{u^4} = u^{3-4} = u^{-1} \) and \( \frac{v^4}{v^7} = v^{4-7} = v^{-3} \). Thus, the expression becomes \( 7 u^{-1} v^{-3} \).

Step 3: Apply the Negative Exponent Outside the Parentheses

The next step is to apply the negative exponent to the entire simplified expression \( (7 u^{-1} v^{-3})^{-1} \). Apply the exponent to each part inside: \( 7^{-1} \), \( u^{1} \), and \( v^{3} \). This results in \( \frac{1}{7} u^{1} v^{3} \).

Step 4: Simplify the Final Expression

Combine all simplified components: The final simplified expression is \( \frac{u v^3}{7} \). This expression contains only positive exponents and is in its simplest form.

Key Concepts

Negative Exponents

Negative exponents might seem intimidating at first, but they're quite straightforward once you understand them. When you see a negative exponent, think of it as a signal that the base needs to be inverted. So, for example, if you have \( x^{-n} \), it is the same as \( \frac{1}{x^n} \). Negative exponents in the numerator can be shifted to the denominator with a positive exponent, and vice versa. This concept greatly helps in simplifying expressions, especially when dealing with fractions.
For instance, in the provided exercise, both \( u^{-1} \) and \( v^{-3} \) arose during simplification. To convert these into positive exponents, you move them to the denominator: \( \frac{1}{u} \) and \( \frac{1}{v^3} \). Understanding how to manipulate negative exponents by converting them is crucial for simplifying algebraic expressions efficiently.

Simplifying Fractions

Simplifying fractions is all about making an expression as simple as possible without changing its value. In algebra, this involves reducing both numerical coefficients and variable parts.
In the exercise, the fraction \( \frac{-49 u^3 v^4}{-7 u^4 v^7} \) is simplified by first canceling out the negative signs because they are in both the numerator and the denominator. This step gives you \( \frac{49 u^3 v^4}{7 u^4 v^7} \).

• The numbers 49 and 7 are simplified to 7 because 49 divided by 7 equals 7. Thus, the coefficient in the fraction simplifies to 7.
• The variable parts \( u^3 \) and \( u^4 \), as well as \( v^4 \) and \( v^7 \), can also be reduced using exponent rules.

The result of these simplifications is a cleaner expression, which allows further manipulation like converting variables with negative exponents.

Exponent Rules

Exponent rules are key when working with algebraic expressions. The most common rules include:

• Product of Powers: If you multiply same bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: If you raise a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product: Distribute the exponent to each factor: \( (ab)^n = a^n b^n \).
• Quotient of Powers: If you divide same bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In the exercise, these rules help simplify \( \frac{u^3}{u^4} \) to \( u^{-1} \) and \( \frac{v^4}{v^7} \) to \( v^{-3} \).Understanding and applying these rules allows us to deal with complex expressions by breaking them down into more manageable parts.

Variables

Variables are symbols that represent numbers and are fundamental in algebraic expressions. They allow for the expression of general mathematical relationships and rules.
In terms of handling variables within fractions and exponents:

• The same base rule is crucial: with \( a^m/a^n = a^{m-n} \), you can easily reduce variables in fractions, provided their exponents are subtracted correctly.
• Sometimes variables have negative exponents, like \( u^{-1} \) or \( v^{-3} \), which can be transitioned to the denominator in a fraction form to become positive. This adjustment is essential for simplifying expressions.

Recognizing how these variable manipulations alter and simplify algebraic expressions is a significant part of solving these problems. By understanding the basic operations with variables, students can tackle even challenging algebraic exercises with confidence.