Problem 43 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \frac{\left(6 y^{3}\right)^{4}}{2 y^{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(648y^7\).

Step 1: Apply the Power Rule

The expression is \( \frac{(6y^3)^4}{2y^5} \). Apply the power rule to \((6y^3)^4\), which states that \((ab)^n = a^n b^n\). So, \((6y^3)^4 = 6^4 (y^3)^4 = 1296 y^{12}\).

Step 2: Substitute into Original Expression

Now replace \((6y^3)^4\) in the original expression with \(1296 y^{12}\), resulting in \(\frac{1296 y^{12}}{2 y^5}\).

Step 3: Simplify the Constants

Divide the constants \(1296\) and \(2\) to get \(648\). The expression now is \(\frac{648 y^{12}}{y^5}\).

Step 4: Apply the Quotient of Powers Rule

Using the quotient of powers rule, \(\frac{a^m}{a^n} = a^{m-n}\), simplify \(\frac{y^{12}}{y^5} = y^{12-5} = y^7\).

Step 5: Write the Final Simplified Expression

Combine the simplified constants and powers to achieve the final expression. The simplified expression is \(648y^7\).

Key Concepts

Power Rule

The power rule is a fundamental concept in exponentiation that simplifies expressions where a power is raised to another power. It states that for any base \(a\) and exponent \(n\), the expression \((a^m)^n\) becomes \(a^{m \cdot n}\). In our original problem, we apply the power rule to \((6y^3)^4\). This breaks down to calculating \(6^4\) and \((y^3)^4\):

• \(6^4\) equals \(1296\), since you multiply 6 by itself four times.
• \((y^3)^4\) becomes \(y^{12}\), by multiplying the exponents 3 and 4.

After using the power rule, the expression \((6y^3)^4\) simplifies to \(1296y^{12}\). This step is crucial because it breaks down complicated exponent expressions into simpler components.

Quotient Rule

The quotient rule helps us simplify expressions involving division of powers with the same base. According to this rule, \(\frac{a^m}{a^n} = a^{m-n}\). This means we subtract the exponent in the denominator from the exponent in the numerator. In step 4 of the problem, we apply this rule to the expression \(\frac{y^{12}}{y^5}\):

• We have the same base \(y\) in both the numerator and the denominator.
• By subtracting the exponents: \(12 - 5 = 7\), we simplify \(\frac{y^{12}}{y^5}\) to \(y^7\).

The quotient rule is essential for decreasing the complexity of fractional exponents, converting them into simpler expressions that are easier to understand.

Simplification

Simplification is the process of reducing an expression to its most concise and understandable form. In our problem, simplification occurs at multiple steps. First, after applying the power rule, we reduce \((6y^3)^4\) to \(1296y^{12}\). Next, we simplify the division of constants:

• From \(\frac{1296}{2}\) to \(648\).

Finally, the expression becomes \(\frac{648y^{12}}{y^5}\), and by applying the quotient rule, it is simplified to \(648y^7\). Simplification is critical because it not only makes expressions smaller but also more practical for further use or interpretation.

Negative Exponents

Dealing with negative exponents is a vital skill in algebra, even if our problem doesn't directly use them. A negative exponent indicates that the base is on the wrong side of a fraction, suggesting it needs to flip over. The rule states that \(a^{-n} = \frac{1}{a^n}\).

• If you encounter a negative exponent, move the base to the other side of the fraction to make the exponent positive.
• For example: \(x^{-3} = \frac{1}{x^3}\).

Although our solution didn't directly incorporate negative exponents, understanding this concept helps prevent errors and enables proper handling of any negative exponents encountered during simplification processes.