Problem 8 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ \frac{95 \sqrt{6}}{9^{1 / 3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{95}{3} \times 3^{1/3} \times \sqrt{6} \).

Step 1: Understand the components

The given expression is \( \frac{95 \sqrt{6}}{9^{1/3}} \). We have a fraction where the numerator is \( 95 \sqrt{6} \) and the denominator is \( 9^{1/3} \), which is the cube root of 9.

Step 2: Simplify the denominator

The denominator \( 9^{1/3} \) can be simplified by recognizing that \( 9 = 3^2 \). Therefore, \( 9^{1/3} = (3^2)^{1/3} = 3^{2/3} \) since the exponents multiply.

Step 3: Combine the terms

Substitute the simplified denominator back into the expression: \( \frac{95 \sqrt{6}}{3^{2/3}} \). This makes it easier to consider multiplying by a form of one to simplify further, if possible.

Step 4: Multiply numerator and denominator by a suitable form of one

To eliminate the fractional exponent in the denominator, multiply the fraction by \( \frac{3^{1/3}}{3^{1/3}} \). This gives \( \frac{95 \sqrt{6} \cdot 3^{1/3}}{3^{2/3} \cdot 3^{1/3}} \).

Step 5: Simplify the new expression

The denominator becomes \( 3^{(2/3)+(1/3)} = 3^1 = 3 \). The numerator simplifies to \( 95 \sqrt{6} \cdot 3^{1/3} \).

Step 6: Write the final simplified form

The expression simplifies to \( \frac{95 \cdot 3^{1/3} \cdot \sqrt{6}}{3} \), which can be rewritten as \( \frac{95}{3} \cdot 3^{1/3} \cdot \sqrt{6} \).

Key Concepts

Fractional Exponents

Fractional exponents may seem daunting at first, but they are quite manageable once you understand the basic principles. In mathematics, fractional exponents represent roots and powers at the same time. For instance, the expression \( a^{m/n} \) is equivalent to \((\sqrt[n]{a})^m\) or \(a^{m}\) raised to the power of \(\frac{1}{n}\). This means:

• The denominator of the fraction (\(n\)) indicates the root.
• The numerator (\(m\)) indicates the power to which the root is raised.

When you encounter an expression like \(9^{1/3}\), you can see that the fraction \(\frac{1}{3}\) denotes that we are seeking the cube root of 9. Remember that expressing numbers with fractional exponents allows us to use algebraic properties to simplify complex expressions easily.

Cube Roots

Cube roots are a particular type of root employed regularly in mathematics. If you are asked to find the cube root of a number \(a\), then you are essentially finding the number \(b\) which, when multiplied by itself three times (\(b \times b \times b\)), gives \(a\). In mathematical terms, this is expressed as \(b = \sqrt[3]{a}\) or \(b = a^{1/3}\).

With cube roots, the exponent \(1/3\) represents the operation to "undo" cubing a number. For instance, if we have \(9 = 3^2\), the cube root of 9 becomes \(9^{1/3} = (3^2)^{1/3} = 3^{2/3}\). This is particularly useful when simplifying expressions with cube roots, as it allows us to manipulate the terms using the properties of exponents.

Simplification Techniques

Simplification techniques in algebra aim to make expressions easier to understand and work with. To simplify expressions containing fractional exponents or roots, we apply a series of algebraic rules. Consider the expression \(\frac{95 \sqrt{6}}{9^{1/3}}\):

• Identify and simplify roots in the denominator. Convert \(9^{1/3}\) into a form \((3^2)^{1/3} = 3^{2/3}\).
• Use the property \(a^m \cdot a^n = a^{m+n}\) to handle and simplify exponents. Multiply both the numerator and denominator by a term designed to "clear" the fractional exponents, like \(\frac{3^{1/3}}{3^{1/3}}\).
• Bring the terms to their simplest form, with whole numbers or simplified radicals whenever possible.

Simplifying expressions effectively allows you to rewrite and understand them better, as seen by expressing our final answer as a more straightforward expression like \(\frac{95}{3} \cdot 3^{1/3} \cdot \sqrt{6}\). This process is crucial in algebra, as it makes solving further equations and making calculations easier and more intuitive.