Problem 41 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{3} b^{-1} z w^{2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^{3} b^{-1} z w^{2}\) using only positive exponents. Answer: \(a^{3} \frac{z w^{2}}{b^1}\)

Step 1: Apply formula to the negative exponent

Identify the negative exponent term, which is \(b^{-1}\). To rewrite this term using a positive exponent, we will apply the formula $$a^{-n} = \frac{1}{a^{n}}$$ to get $$b^{-1} = \frac{1}{b^1}$$.

Step 2: Rewrite the expression with positive exponent

Replace the negative exponent term in the original expression with the positive exponent term. So the expression becomes: $$a^{3} b^{-1} z w^{2} = a^{3} \left(\frac{1}{b^{1}}\right) z w^{2}$$

Step 3: Simplify the expression

Since we now have the whole expression with only positive exponents, we can simplify it as: $$a^{3} \frac{z w^{2}}{b^1}$$ Now we have successfully rewritten the original expression using only positive exponents: $$a^{3} b^{-1} z w^{2} = a^{3} \frac{z w^{2}}{b^1}$$

Key Concepts

Positive Exponents

Understanding positive exponents is essential in algebra. When you see a positive exponent, it simply means you multiply the base by itself as many times as the exponent indicates. For example,

• \( a^3 \) means \( a \times a \times a \).

Positive exponents express repeated multiplication, which is straightforward when you remember this rule.
Whenever working with expressions involving positive exponents, simply multiply the base repeatedly according to the exponent given. This makes computations easier and faster by organizing repeated multiplications into a single term with a clearer meaning.

Negative Exponents

Negative exponents can be confusing at first, but they have a clear meaning in mathematics. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In other words,

• \( b^{-1} \) becomes \( \frac{1}{b^1} \).

This transformation changes a negative exponent into a positive one, making algebraic expressions easier to work with.
Converting negative exponents to positive helps simplify expressions and solve equations. Always remember, when you encounter a term with a negative exponent, flip the base into its reciprocal form and change the exponent to positive to simplify the expression.

Exponent Rules

Exponent rules are essential for simplifying and manipulating algebraic expressions. These rules help in streamlining complex calculations and solving various types of equations. Some basic exponent rules include:

• The Product of Powers Rule: \( a^m \times a^n = a^{m+n} \)
• The Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)
• The Power of a Product Rule: \( (ab)^n = a^n \times b^n \)

By combining these rules with the knowledge of positive and negative exponents, you can effectively simplify algebraic expressions.
Using these rules, complex expressions become manageable because they allow for step-by-step simplification to reach a solution without errors. Mastery of these exponent rules equips you with the tools needed to tackle a wide variety of algebraic problems.