Problem 56 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(a^{5} b^{-5} c^{-4}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{b^{15} c^{12}}{a^{15}} \).

Step 1: Understanding Negative Exponents

In the expression \((a^{5}b^{-5}c^{-4})^{-3}\), notice that \(-3\) is outside the parentheses, affecting each term within the brackets. It is important to know that a negative exponent represents the reciprocal of the base raised to the opposite positive power. For a compound expression, distributing the negative exponent to each component will yield positive exponents.

Step 2: Applying the Negative Exponent

We begin by applying the exponent \(-3\) to each factor within the parenthesis:\[(a^{5}b^{-5}c^{-4})^{-3} = rac{1}{(a^{5}b^{-5}c^{-4})^{3}}\]This step moves every term within the brackets to the denominator with the positive power of \(3\).

Step 3: Distributing the Power

Distribute the power of \(3\) across each variable:\[rac{1}{a^{5 imes 3}b^{-5 imes 3}c^{-4 imes 3}}\]Simplifying the exponents gives:\[rac{1}{a^{15}b^{-15}c^{-12}}\]

Step 4: Converting Negative Exponents

To express the expression without negative exponents, move any term with a negative exponent from the denominator to the numerator, inverting their powers from negative to positive. Thus, we have:\[a^{-15}b^{15}c^{12}\]

Step 5: Final Simplified Expression

Combine all terms with positive exponents in the numerator, yielding:\[b^{15} c^{12} a^{-15}\]To express only positive exponents correctly in standard form, the expression becomes:\[\frac{b^{15} c^{12}}{a^{15}}\]

Key Concepts

Negative Exponents

Negative exponents might look a bit scary, but they're just another way of talking about fractions! When you see a negative exponent, it simply means you're dealing with the reciprocal of the base. The reciprocal of a number - is what you get when you divide 1 by that number. For instance, having,

• an exponent of
  
•