Problem 69 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{8 \sqrt[3]{54 m^{7}}}{\sqrt[3]{2 m}}\)

Step-by-Step Solution

Answer

The simplified form is \( 24 m^{2} \).

Step 1: Identify the Problem

We need to simplify the expression \( \frac{8 \sqrt[3]{54 m^{7}}}{\sqrt[3]{2 m}} \) using the properties of exponents and radicals.

Step 2: Apply the Quotient Rule for Radicals

Recall that for any cube roots \( \sqrt[3]{a} \) and \( \sqrt[3]{b} \), \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} \). Use this to rewrite the expression as \( 8 \sqrt[3]{\frac{54 m^{7}}{2 m}} \).

Step 3: Simplify Inside the Radical

Reduce the fraction inside the cube root: \( \frac{54 m^{7}}{2 m} = \frac{54}{2} \cdot \frac{m^{7}}{m} = 27 m^{6} \).

Step 4: Evaluate the Cube Root

Find \( \sqrt[3]{27 m^{6}} \). Since \( \sqrt[3]{27} = 3 \) and \( \sqrt[3]{m^{6}} = m^{2} \), it follows that \( \sqrt[3]{27 m^{6}} = 3 m^{2} \).

Step 5: Combine and Simplify Further

Multiply the result from the cube root by 8 as per the original expression: \( 8 \times 3 m^{2} = 24 m^{2} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves making expressions containing radicals easier to work with. Radicals often appear as square roots, cube roots, or higher roots. The key to simplifying them is to break down the number or expression under the radical sign into smaller parts that have exact roots.

• For example, to simplify \(\sqrt[3]{54}\), find the prime factors: \(54 = 2 \times 3^3\). Since 3 cubed is a perfect cube, \(\sqrt[3]{54} = 3 \sqrt[3]{2}\).

With algebraic expressions inside radicals, apply the same method. Identify parts of the expression you can take out from under the radical by recognizing patterns or using factorization.
This is a practical step in both simplifying individual components and making things more manageable for further mathematical operations.

Cube Roots

Cube roots address finding a number which when multiplied by itself three times gives the original number. The cube root is represented by the radical symbol with a small 3 at its "hook": \(\sqrt[3]{a}\).

• Taking the cube root of a perfect cube like \(27\) is straightforward: since \(27 = 3 \times 3 \times 3\), \(\sqrt[3]{27} = 3\).

Cube roots are not limited to just numbers but also apply to variable expressions, as demonstrated in \(\sqrt[3]{m^6}\). Express \(m^6\) as \((m^2)^3\), leading to \(\sqrt[3]{m^6} = m^2\).
Understanding cube roots aids in simplifying complex algebraic expressions, especially when reducing terms inside a radical for operations such as division or multiplication.

Properties of Exponents

Exponents are a shorthand to represent repeated multiplication of a number by itself. They follow specific rules that ease solving algebraic expressions. Here are a few essential properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)

In the original exercise, you can simplify \(\frac{m^7}{m}\) using quotient of powers: \(m^{7-1} = m^6\).
Mastering these rules helps in managing and simplifying expressions, especially when dealing with algebraic fractions and radicals.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but consist of polynomials in the numerator, denominator, or both. Simplifying them involves factoring and reducing just like with numerical fractions.
A few steps to consider when simplifying algebraic fractions include:

• Factor the numerator and the denominator.
• Cancel any common factors between them.

In the exercise, reducing \(\frac{54}{2}\) to \(27\) is a step of simplifying such an algebraic fraction. This also involves recognizing when you can apply these fractions within radicals before using further rules like the properties of exponents.
These operations make algebraic expressions neater, minimizing their complexities, which is vital for solving equations and performing more advanced mathematics.