Problem 56 - Elementary Algebra

Question

Write \(4 a^{-6} b^{2} c^{3} a^{5} b^{-3}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Simplify the expression \(4a^{-6}b^{2}c^{3}a^{5}b^{-3}\) and rewrite it using only positive exponents. Answer: \(\dfrac{4c^3}{ab}\)

Step 1: Identify the bases with the same exponents and apply the exponent rule.

The given expression is: \(4a^{-6}b^{2}c^{3}a^{5}b^{-3}\). We can see that there are two 'a' terms, \(a^{-6}\) and \(a^{5}\), and two 'b' terms, \(b^{2}\) and \(b^{-3}\). We can apply the rule \(a^{m} \cdot a^{n} = a^{m+n}\) for these terms.

Step 2: Combine the 'a' terms and the 'b' terms.

Combine the 'a' terms: \(a^{-6} \cdot a^{5} = a^{-6+5} = a^{-1}\) Combine the 'b' terms: \(b^{2} \cdot b^{-3} = b^{2-3} = b^{-1}\) Now, the expression becomes: \(4a^{-1}b^{-1}c^{3}\).

Step 3: Rewrite the expression using only positive exponents.

Use the rule \(a^{-n} = \dfrac{1}{a^n}\) to rewrite the expression with positive exponents: \(a^{-1} = \dfrac{1}{a}\) \(b^{-1} = \dfrac{1}{b}\) Substitute these into the expression: \(4a^{-1}b^{-1}c^{3} = 4\cdot\dfrac{1}{a}\cdot\dfrac{1}{b}\cdot c^{3}\)

Step 4: Simplify the expression.

Combine the fractions in the expression: \(4\cdot\dfrac{1}{a}\cdot\dfrac{1}{b}\cdot c^{3} = 4\cdot\dfrac{1}{ab}\cdot c^{3}\). And finally, rewrite the expression as \(\dfrac{4c^3}{ab}\), where all exponents are positive.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial in algebra as it simplifies expressions with powers into easier forms. One fundamental rule is the Product of Powers, which states that when multiplying two powers with the same base, you can add the exponents: \( a^m \cdot a^n = a^{m+n} \). Simply put, instead of multiplying the base several times, you can perform a quicker addition of exponents.

Another important rule is for dealing with negative exponents. The rule states \( a^{-n} = \frac{1}{a^n} \), which means a negative exponent indicates a reciprocal of the base raised to the positive opposite of that exponent. This transformation allows the expression to have only positive exponents, aligning with the standard form in algebra. Using these rules appropriately is essential for not only simplifying algebraic expressions but also for laying the groundwork for more complex operations in algebra.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing complexity while maintaining the original value of the expression. It generally involves applying the exponent rules we've discussed, distributing properties, combining like terms, and factoring, when necessary. For instance, when you see terms with the same base being multiplied, like in our example \( a^{-6} \cdot a^{5} \), they can be simplified by adding the exponents to get a single term with that base, \( a^{-1} \).

Then, if negative exponents appear, you convert them to positive by taking the reciprocal, which often results in an algebraic fraction. Simplifying not only makes expressions cleaner and easier to work with but also prepares them for further operations like solving equations or inequalities.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. Simplifying these fractions is similar to simplifying numerical fractions: look for common factors in the numerator and denominator and cancel them out. In our exercise, we encounter negative exponents, which lead us to algebraic fractions when rewritten with positive exponents.

It's important to rewrite expressions with negative exponents to positive ones in the fractions to make them easier to interpret and calculate with. For example, \( a^{-1} \) becomes \( \frac{1}{a} \), and \( b^{-1} \) becomes \( \frac{1}{b} \). In the end, we may combine these algebraic fractions by finding a common denominator, which allows us to add or subtract fractions or simply to represent them in a simplified form, like \( \frac{4c^3}{ab} \), ensuring all variables have positive exponents. Such representations are essential for further algebraic manipulations.