Problem 55 - Algebra A Combined Function

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 Power of a Power Rule

The expression \((3x^{1/4})^3\) can be simplified using the power of a power rule, which states that \((a^m)^n = a^{m \times n}\).Apply this rule: \[(3x^{1/4})^3 = 3^3 \cdot (x^{1/4})^3 = 27 x^{3/4}\]

Step 2: Divide Using the Quotient Rule

Next, simplify the division of exponents using the quotient rule, which states \( \frac{a^m}{a^n} = a^{m-n} \).Subtract the exponent in the denominator from the exponent in the numerator: \[\frac{27 x^{3/4}}{x^{1/12}} = 27 x^{(3/4 - 1/12)}\]

Step 3: Simplify the Exponent

Calculate the expression \(3/4 - 1/12\) to simplify the exponent of \(x\).Find a common denominator:- \(3/4 = 9/12\), so the operation becomes \(9/12 - 1/12\).- Simplify: \(9/12 - 1/12 = 8/12 = 2/3\).Thus, the expression simplifies to:\[27 x^{2/3}\]

Step 4: Conclusion

Now that we have simplified the expression with all positive exponents, the final answer is:\[27 x^{2/3}\]

Key Concepts

Power of a Power Rule

When you encounter an expression like \((a^m)^n\), it's time to use the power of a power rule. This rule helps in simplifying expressions where an exponent is raised to another exponent. The trick here is to multiply the exponents together. So, the rule states that \((a^m)^n = a^{m \times n}\).

Let's look at how this applies to our example: \((3x^{1/4})^3\). Here, you are raising the entire bracket to the power of 3. Use the rule by distributing the power of 3 to both the constant 3 and the term with \(x\).

First, apply the exponent to the constant: \(3^3 = 27\).
Next, multiply the exponents of \(x\): \((x^{1/4})^3 = x^{1/4 \times 3} = x^{3/4}\).

An expression \((3x^{1/4})^3\) simplifies to \(27x^{3/4}\) using this rule!
This is useful whenever you need to "unpack" complex exponential terms.

Quotient Rule

Dealing with division of powers can be easy if you know the quotient rule. This rule is your friend when you want to simplify expressions like \(\frac{a^m}{a^n}\). It tells you to subtract the exponent in the denominator from the exponent in the numerator: \(a^{m-n}\).

In our problem, we have the division \(\frac{27x^{3/4}}{x^{1/12}}\). Here, the quotient rule will simplify the powers of \(x\).

Let's simplify this step-by-step:

• Keep the constant 27 as it is, because only similar bases are affected by exponent rules.
• Subtract the exponents: \(\frac{x^{3/4}}{x^{1/12}} = x^{3/4 - 1/12}\).

This subtraction might need finding a common denominator. We'll explore that next. The quotient rule brings order to potential chaos in division scenarios!

Simplifying Expressions

Simplifying expressions often involves finding common denominators when working with fractions. When we have exponents like \(\frac{3}{4}\) and \(\frac{1}{12}\), we need a common baseline for subtraction. This step makes life easier when combining or subtracting fractional exponents.

Focus on these simple steps to achieve this:

• Convert \(\frac{3}{4}\) to an equivalent fraction with the same denominator as \(\frac{1}{12}\).
• Multiply \(\frac{3}{4}\) by 3 to get \(\frac{9}{12}\).
• Next, perform the subtraction: \(\frac{9}{12} - \frac{1}{12} = \frac{8}{12}\).
• Finally, simplify \(\frac{8}{12}\) to \(\frac{2}{3}\) by dividing both top and bottom by 4.

This transforms the power of \(x\) to \(x^{2/3}\).
In the end, combining what we have done yields the simplified expression: \(27 x^{2/3}\).
Always remember these steps to make simplifying a smooth process!