Problem 55 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(3 x^{1 / 4}\right)^{3}}{x^{1 / 12}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 27x^{2/3} \).

Step 1: Apply the Power Rule

The given expression is \( \frac{(3x^{1/4})^3}{x^{1/12}} \). First, apply the power rule \( (a^m)^n = a^{m \cdot n} \) to the numerator \( (3x^{1/4})^3 \). This results in \( 3^3 \times (x^{1/4})^3 \), which simplifies to \( 27 \times x^{3/4} \).

Step 2: Apply the Quotient Rule

Use the quotient rule of exponents \( \frac{a^m}{a^n} = a^{m-n} \) to simplify \( \frac{x^{3/4}}{x^{1/12}} \). Subtract the exponents: \( x^{3/4 - 1/12} \). To subtract, convert the exponents to a common denominator: \( 3/4 = 9/12 \). Therefore, the expression is reduced to \( x^{9/12 - 1/12} = x^{8/12} \).

Step 3: Simplify the Exponent

Now, simplify the exponent \( 8/12 \) by reducing the fraction. Divide both the numerator and the denominator by 4 to get \( 2/3 \). Thus, the expression becomes \( x^{2/3} \).

Step 4: Combine the Simplified Expression

Combine the results from the simplified powers. The overall expression is \( 27 \times x^{2/3} \). Ensure that each part of the expression is expressed with positive exponents.

Key Concepts

Power Rule

The Power Rule is an essential tool in simplifying expressions with exponents. It states that when you have a power raised to another power, you multiply the exponents together. For example, in the expression \((a^m)^n\), you would apply the power rule to get \(a^{m \cdot n}\). In our original exercise, the expression \((3x^{1/4})^3\) uses the power rule. Each component inside the parentheses needs to be raised to the power of 3.

• The constant 3 is raised to the power of 3, resulting in \(3^3 = 27\).
• The variable \(x^{1/4}\) is raised to the power of 3, resulting in \(x^{1/4 \cdot 3} = x^{3/4}\).

Together, this results in \(27 \times x^{3/4}\) after applying the power rule.

Quotient Rule

The Quotient Rule helps us when dividing expressions with the same base. It tells us to subtract the exponents from one another. The rule is \(\frac{a^m}{a^n} = a^{m-n}\), where you simplify by subtracting the exponent in the denominator from the exponent in the numerator.Applying the quotient rule to \(\frac{x^{3/4}}{x^{1/12}}\) involves the subtraction of the two exponents.

• Firstly, we convert \(3/4\) to a common denominator with \(1/12\), which is 12. This turns \(3/4\) into \(9/12\).
• Then, subtract \(1/12\) from \(9/12\) to get \(8/12\).

Thus, the simplified expression becomes \(x^{8/12}\), which we simplify further by reducing the fraction.

Positive Exponents

Positive exponents are crucial for expressing our final answer in a simplified and standard form. An exponent indicates how many times to use the base in a multiplication. A positive exponent means this number is to be repeated as a factor.The step to positive exponents ensures that answers follow convention and reduce confusion.

• After simplifying with the quotient rule, we had \(x^{8/12}\), a fraction that can be simplified further.
• Simplification of \(8/12\) involves reducing it to \(2/3\) by dividing both the numerator and the denominator by 4.
• This gives us \(x^{2/3}\), which is expressed as a positive exponent.

Our final simplified expression is \(27 \times x^{2/3}\), all expressed in positive exponents for clarity and standardization.