Problem 18 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{3 a^{5} b^{3} c^{3}}{9 a^{3} b^{7} c} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{a^{2}c^{2}}{3b^{4}} \).

Step 1: Split the Fraction

Start by expressing the fraction \[ \frac{3 a^{5} b^{3} c^{3}}{9 a^{3} b^{7} c} \]as separate fractions for each variable: \[ \frac{3}{9} \times \frac{a^{5}}{a^{3}} \times \frac{b^{3}}{b^{7}} \times \frac{c^{3}}{c} \]

Step 2: Simplify the Constants

Divide the constants 3 and 9. This gives:\[ \frac{3}{9} = \frac{1}{3} \]

Step 3: Simplify the Powers of 'a'

Apply the power law for exponents, which states that \( x^m / x^n = x^{m-n} \). Thus,\[ \frac{a^{5}}{a^{3}} = a^{5-3} = a^{2} \]

Step 4: Simplify the Powers of 'b'

Apply the same power law for the base 'b'. Thus,\[ \frac{b^{3}}{b^{7}} = b^{3-7} = b^{-4} = \frac{1}{b^4} \]

Step 5: Simplify the Powers of 'c'

Similarly for the base 'c', apply the power law:\[ \frac{c^{3}}{c} = c^{3-1} = c^{2} \]

Step 6: Reconstruct the Expression

Combine all the simplified fractions:\[ \frac{1}{3} \times a^{2} \times \frac{1}{b^{4}} \times c^{2} = \frac{a^{2}c^{2}}{3b^{4}} \]

Key Concepts

Constants in Algebra

In algebra, constants are fixed values that do not change. These could be numbers like 2, 5, or in our case, 3 and 9. When simplifying equations, handling constants is usually straightforward. They can be combined or divided just like ordinary numbers. In the given problem, we had to simplify the fraction \( \frac{3}{9} \). This is simply a matter of dividing both 3 and 9 by their greatest common divisor, which in this case is 3. So, \( \frac{3}{9} \) simplifies to \( \frac{1}{3} \). By reducing the constants, we make the entire expression easier to work with.

• Constants provide the numeric value in an expression.
• Always look to simplify constants first, if possible.
• They follow regular arithmetic rules.

By simplifying constants at the outset, you clear clutter from your expression right away, making later algebraic operations, like dealing with variables and exponents, more manageable.

Power Law for Exponents

The power law for exponents is an essential tool in algebra. When you have the same base raised to different powers and you're dividing them, you subtract the exponents. This rule can be written as \( x^m / x^n = x^{m-n} \).
Let's look at how this rule was used in the problem:

• For the variable \( a \), we had \( a^5 / a^3 \). By applying the power law, we get \( a^{5-3} = a^2 \).
• For the variable \( b \), \( b^3 / b^7 \) became \( b^{3-7} = b^{-4} \), which simplifies to \( \frac{1}{b^4} \).
• Finally, for \( c \),\( c^3 / c \) simplifies to \( c^{3-1} = c^2 \).

These applications make fractional expressions simpler and more manageable. Understanding how to utilize the power law for exponents is key to working efficiently with algebraic expressions involving powers.

Expression Reconstruction

Expression reconstruction is the final step in simplifying an algebraic fraction. Once individual components are simplified, they must be pieced back together. In our example, each variable's fraction was simplified, and the constants were reduced.
Let's see how it all came together:

• After simplifying, we had \( \frac{1}{3} \) for the constants, \( a^2 \) for \( a \), \( \frac{1}{b^4} \) for \( b \), and \( c^2 \) for \( c \).
• These are multiplied together: \( \frac{1}{3} \times a^2 \times \frac{1}{b^4} \times c^2 \).
• This results in the final simplified fraction: \( \frac{a^2 c^2}{3b^4} \).

Expression reconstruction ensures that all parts are correctly compiled to reflect the original equation's relationships. The key is to maintain proper order and grouping so that nothing is lost or misrepresented in the simplification process. This reconciles individual components into a coherent, simplified form of the original expression.