Problem 71 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{q^{-1} r^{-1} s^{-2}}{r^{-5} s q^{-8}}\right)^{-1} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{q^9 r^6 s}\).

Step 1: Recognize and Rewrite the Negative Exponent

The given expression is: \[\left(\frac{q^{-1} r^{-1} s^{-2}}{r^{-5} s q^{-8}}\right)^{-1}\]To eliminate the negative exponent on the outside of the fraction, we can take the reciprocal of the fraction inside the parenthesis. This means we swap the numerator and denominator, flipping all the exponents to positive:\[\frac{r^{-5} s q^{-8}}{q^{-1} r^{-1} s^{-2}} \]

Step 2: Simplify the Numerator

Focus on the numerator:- \(r^{-5}\) becomes \(\frac{1}{r^5}\) when moved to the denominator.- \(s\) remains in the numerator as \(s^1\).- \(q^{-8}\) becomes \(\frac{1}{q^8}\) when moved to the denominator.This gives us:\[\frac{s}{q^8 r^5} \]

Step 3: Simplify the Denominator

Focus on the denominator:- \(q^{-1}\) becomes \(q\) when moved to the numerator.- \(r^{-1}\) becomes \(r\) when moved to the numerator.- \(s^{-2}\) becomes \(s^2\) when moved to the numerator.This gives us:\[q r s^2\]

Step 4: Combine and Simplify Final Expression

Now we substitute the simplified numerator and denominator back:\[\frac{s}{q^8 r^5} \div q r s^2 = \frac{s}{q^8 r^5} \times \frac{1}{q r s^2}\]Combine all terms:- Numerator: \(s\)- Denominator: \(q^9 r^6 s^2\)So the expression simplifies to:\[\frac{1}{q^9 r^6 s}\]

Step 5: Final Expression

After simplification, the expression is:\[\frac{1}{q^9 r^6 s}\]This has accounted for all negative exponents by moving terms to their appropriate positions in the fraction with positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem confusing at first, but once you understand the concept, they become much easier to work with. When a number or variable has a negative exponent, it means we are dealing with a reciprocal--a fancy word for flipping things around to the denominator or numerator.

• For example, if you have a term like \(x^{-n}\), you can rewrite it as \(\frac{1}{x^n}\). This shows us that negative exponents simply mean "one over" the base raised to the positive exponent.
• This also means that if we encounter a negative exponent in the denominator, such as \(\frac{1}{y^{-n}}\), we can move it to the numerator as \(y^n\).

Most importantly, the rule also implies that any term can be flipped; so when combined with fraction operations, it can simplify an otherwise complex expression.

Fraction Simplification

Simplifying fractions is all about breaking down a complicated fraction into a more understandable form. We do this by canceling out common terms in the numerator and the denominator. In algebra, this might involve variables with exponents rather than just numbers.

• First, identify common bases in both the numerator and the denominator, these could be variables like \(q\), \(r\), or \(s\) in the original exercise.
• Next, apply the rules of exponents: subtract the exponents when dividing like bases (i.e., \(a^m/a^n = a^{m-n}\)).

Fraction simplification often involves moving components with negative exponents, which allows them to combine easily with other like terms through subtraction or addition, leading to a much simpler form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Solving them involves understanding the properties of exponents, fractions, and mathematical operations. Let's examine a few key ideas:

• Consistency is key: Always apply the same rules and methods to each part of the expression.
• Understand the hierarchy: Operations inside brackets and those above or below a fraction line often take precedence.
• Exponent rules simplify expressions: Particularly the rules of adding, subtracting, and handling negative exponents or powers of a product.

By manipulating and simplifying these expressions carefully, you’ll be able to reach solutions efficiently and accurately. Algebraic expressions may look daunting, but with these tools, they become manageable.