Problem 51 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ \frac{1}{2 x^{8} y^{-5}} $$

Step-by-Step Solution

Answer

The expression \(\frac{1}{2 x^{8} y^{-5}}\) rewritten with positive exponents is \(\frac{y^{5}}{2 x^{8}}\).

Step 1: Identify Negative Exponents

Starting with the expression \(\frac{1}{2 x^{8} y^{-5}}\), it's noticeable that the \(y\) term has a negative exponent: \(-5\).

Step 2: Apply Rule for Negative Exponents

The property \(a^{-n} = 1 / a^n\) can be applied to rewrite the expression with positive exponents by moving the base \(y\) with the negative exponent to the top of the fraction, yielding the result \(\frac{1}{2 x^{8}} * y^{5}\).

Step 3: Simplify Expression

To simplify the expression further, one can transform the multiplication of fractions into a single fraction. This step leads to the final expression \( \frac{y^{5}}{2 x^{8}} \)

Key Concepts

Negative Exponents

In algebra, you often encounter negative exponents. A negative exponent tells you how many times to divide by the base, as opposed to multiplying. The rule to remember is:

• For any nonzero number, the expression \(a^{-n}\) equals \(1/a^n\).

When you see a term like \(y^{-5}\), it means "take the reciprocal" of \(y\) to the power of 5. Instead of multiplying \(y\) five times, you're dividing by \(y\) five times.
To make it positive, simply "flip" it to the opposite part of the fraction - from the denominator to the numerator, or vice versa.
In the expression \(\frac{1}{2x^8y^{-5}}\), we move the \(y^{-5}\) up to become \(y^5\) in the numerator. This use of negative exponents is crucial for simplifying and writing expressions in a standard positive form.

Positive Exponents

When exponents are positive, they describe repeated multiplication. For example, \(x^8\) means you multiply \(x\) by itself 8 times (\(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\)).
Positive exponents are straightforward; they show how many times you multiply the base.

• An expression like \(2x^8\) means 2 times \(x\) raised to the eighth power.
• Keeping all exponents positive gives a cleaner, easier-to-read form.

In our example, \(y^5\) is now in the numerator, and \(x^8\) remains in the denominator. This clearly shows the product of \(y\) five times and division by \(x\) eight times.
Understanding positive exponents helps to correctly interpret the magnitude and relationship of values in algebraic expressions.

Simplifying Expressions

Simplifying expressions is the process of making an equation or expression easier to read or solve. It involves reducing the expression to its simplest form while keeping equality intact.
The ultimate goal is to have a neat "package" that communicates the same value with less complexity. It often results in converting division and multiplication from separate terms to a single, cohesive fraction.

• In our example, after moving the term \(y^5\) to the numerator, we have \(\frac{y^5}{2x^8}\).
• This step implements all basic rules: shifting any negative exponents to positive and organizing the expression.

Simplifying does not alter the expression's value but makes calculations and understanding more straightforward.
With practice, simplifying algebraic expressions becomes a powerful tool to solve equations effortlessly.