Problem 52 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(\frac{a^{5} b c^{0}}{a^{7} b^{-2}}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a^6}{b^9} \).

Step 1: Cancel out common terms within the fraction

The expression is \( \frac{a^{5} b c^{0}}{a^{7} b^{-2}} \). Since \( c^0 = 1 \), we can simplify it without changing the value. Also, simplify \( b \) terms: \( b = b^1 \). Therefore, the expression simplifies to \( \frac{a^5 \, b^1}{a^7 \, b^{-2}} \), which can be further simplified by reducing terms to \( a^{5-7} \, b^{1+2} = a^{-2} \, b^{3} \).

Step 2: Apply the Negative Exponent

Next, we incorporate the exponent outside the fraction: \( \left(a^{-2} \, b^{3}\right)^{-3} \). When raising a power to another exponent, multiply the exponents: \( a^{-2 \times -3} \cdot b^{3 \times -3} = a^{6} b^{-9} \).

Step 3: Express Using Positive Exponents

Convert any negative exponents into positive by taking the reciprocal. Thus, \( b^{-9} \) becomes \( \frac{1}{b^{9}} \). The final expression is \( a^6 \cdot \frac{1}{b^9} = \frac{a^6}{b^9} \).

Key Concepts

Negative Exponents

Understanding negative exponents can be quite simple once you get the hang of it. At its core, a negative exponent tells you how many times to divide by the number, instead of multiplying. A negative exponent doesn’t mean you have a negative number or expression, but rather the reciprocal of the base raised to the opposite positive power. For instance,

• If you have \( a^{-n} \), it implies \( \frac{1}{a^n} \). So, it flips the base to its reciprocal.

This technique is vital in simplifying expressions with negative exponents, as you systematically rewrite them with positive exponents after simplification. For example, in our step-by-step solution, we encountered \( b^{-9} \) which was converted to \( \frac{1}{b^9} \), effectively making the expression more manageable. This change does not alter the numeric value, it merely changes the form into something easier to compute or visualize.

Exponent Rules

Exponent rules form the backbone of simplifying algebraic expressions involving powers. Learning to apply these rules can make operations with exponents much faster and easier. Here are some of the key rules:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \). Simply add the exponents when multiplying like bases.
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \). Subtract the exponents when dividing like bases.
• Power of a Power Rule: \( (a^m)^n = a^{m \times n} \). Multiply the exponents when raising a power to another power.

In the provided solution, we used several of these rules. For example, when simplifying the expression \( \frac{a^5 \, b}{a^7 \, b^{-2}} \), we applied the quotient of powers rule to get \( a^{-2} \, b^3 \). Also, raising \( a^{-2} \, b^3 \) to the power of \(-3\) involved the power of a power rule: \( a^{-2 \times -3} \cdot b^{3 \times -3} = a^6 b^{-9} \). Thorough practice with these rules will simplify your work with exponents.

Fractional Exponents

Fractional exponents might initially seem more complicated than they really are. A fractional exponent, such as \( a^{1/n} \), means the \( n \)-th root of \( a \). To illustrate:

• If you have \( a^{1/2} \), it is equivalent to \( \sqrt{a} \), or the square root of \( a \).
• Similarly, \( a^{3/4} \) is \( (\sqrt[4]{a^3}) \), or \( a space ext{1/4th powered and then cubed} \).

By converting between roots and fractional exponents, you gain additional tools for simplifying or manipulating expressions where traditional powers might not suffice. While the original exercise didn’t directly involve fractional exponents, understanding them deepens your ability to handle various types of exponents, especially when combined in more advanced algebraic contexts. Practice observing how these relationships work enhances your fluency with exponents in general.