Problem 140 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(\frac{5^{-1} a^{3} b^{-6}}{x^{-2} y^{9}}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with only positive exponents: $\left(\frac{5^{-1} a^{3} b^{-6}}{x^{-2} y^{9}}\right)^{2}$ Answer: $\frac{25a^6x^4}{b^{12}y^{18}}$

Step 1: Distribute the power to each term inside the parentheses

Start by distributing the power of 2 to every term inside the parentheses. Apply the power on each term by multiplying the exponents. $$ \left(\frac{5^{-1} a^{3} b^{-6}}{x^{-2} y^{9}}\right)^{2} = \frac{(5^{-1})^2 (a^3)^2 (b^{-6})^2}{(x^{-2})^2 (y^9)^2} $$

Step 2: Simplify the exponents

Calculate the new exponents by multiplying them with the power of 2, such that negative exponents become positive and positive exponents get squared. $$ = \frac{5^{-2} a^6 b^{-12}}{x^{-4} y^{18}} $$

Step 3: Move the terms with negative exponents

To make sure only positive exponents appear, we need to move the terms with negative exponents to the other side of the fraction (from numerator to denominator and vice versa). Apply this rule: $$ \frac{a^{-n}}{b^{-m}} = \frac{b^m}{a^n} $$ $$ = \frac{5^2 a^6 x^4}{b^{12} y^{18}} $$

Step 4: Simplify the final expression

Now that all exponents are positive, simplify the final expression. $$ = \frac{25a^6x^4}{b^{12}y^{18}} $$ The desired expression with only positive exponents is: $$ \frac{25a^6x^4}{b^{12}y^{18}} $$

Key Concepts

Simplifying Expressions

When you simplify expressions, you aim to create a more concise form of the expression without changing its value. It's like tidying up a messy room: you still have everything you need, just organized differently. Simplification involves:

• Combining like terms
• Applying mathematical rules, such as distributing exponents
• Moving terms to ensure all have positive exponents

In our exercise, simplifying involves distributing the power of 2 to each term in the expression.
This process helps in transforming complex-looking expressions into simpler, more manageable forms without altering their inherent value.

Algebraic Fractions

Algebraic fractions are like regular fractions but with variables. These expressions play a crucial role in algebra and require careful handling of both numerators and denominators. It's essential to remember:

• The same rules for fractions apply, but we also manipulate variables.
• Operations include simplifying, multiplying, or dividing by moving terms between the numerator and the denominator.
• It's key to convert negative exponents to positive, which often involves moving the base and exponent across the fraction bar.

In our example, notice how variables with negative exponents are shifted. As a result, those terms move from numerator to denominator or vice versa, allowing us to keep exponents positive.

Negative Exponents

Negative exponents can be confusing, but understanding their rules is crucial. A term with a negative exponent indicates division, instead of multiplication, as it would with positive exponents. Key rules include:

• A negative exponent indicates the reciprocal. For example, for any non-zero number \(a^{-n} = \frac{1}{a^n}\).
• To make exponents positive, you can "flip" the term across the fraction line.
• When simplifying, watch for negative exponents and move the term to the other side of the fraction to make them positive.

In the given exercise, applying this rule helps turn negative exponents into positive ones by switching their placement in the fraction. This adjustment is crucial for simplifying and ensuring only positive exponents remain.