Problem 51 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt[3]{189 a^{5}}}{\sqrt[3]{7 a}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3a^{4/3} \).

Step 1: Simplify the Radicals

The expression given is \( \frac{\sqrt[3]{189 a^{5}}}{\sqrt[3]{7 a}} \). Start by expressing the radicals as a single radical. This gives us \( \sqrt[3]{\frac{189 a^5}{7 a}} \).

Step 2: Simplify the Fraction Inside the Radical

Simplify the fraction \( \frac{189 a^5}{7 a} \) by dividing both the numerator and the denominator by \(7a\). This results in \( \frac{189}{7} = 27 \) and \( \frac{a^5}{a} = a^4 \). Thus, the expression under the radical becomes \( 27 a^4 \).

Step 3: Calculate the Cube Root

Now, calculate \( \sqrt[3]{27 a^4} \). Since \( 27 = 3^3 \), the cube root of 27 is 3. Similarly, for \( a^4 \), \( \sqrt[3]{a^4} = a^{4/3} \). This gives us the simplified form \( 3a^{4/3} \).

Key Concepts

Understanding Cube Roots

Cube roots are a type of radical expression where we aim to find a number that, when multiplied by itself three times, produces the original number. For example, the cube root of 8 is 2 because

• 2 x 2 x 2 = 8.

To express cube roots in terms of algebra, we use the notation

• \( \sqrt[3]{x} \), which represents the cube root of \( x \).

In the original exercise, we are tasked with simplifying \( \sqrt[3]{189 a^{5}} \) and \( \sqrt[3]{7 a} \). To ease simplification, these two terms were combined into a single cube root:

• \( \sqrt[3]{\frac{189 a^5}{7 a}} \).

This step helps us to manage the expression inside the cube root more easily before finding the individual components' cube roots.

Simplifying Fractions Inside Radicals

Fraction simplification is crucial, especially when dealing with radical expressions. It involves breaking down a fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and the denominator. In our problem, the fraction \( \frac{189 a^5}{7 a} \) appears inside the cube root.
We simplify it as follows:

• Divide 189 by 7, which gives 27 (since 189 ÷ 7 = 27).
• Divide \( a^5 \) by \( a \), yielding \( a^4 \).

Thus, the simplified fraction inside the cube root is \( 27 a^4 \). By breaking down the expression inside the radical, we make it easier to calculate the cube root in the next step.

Simplifying Radical Expressions

Radical expressions often involve taking roots of numbers or variables, and in this exercise, we are dealing with both.
When simplifying \( \sqrt[3]{27 a^4} \), we're tasked with finding the cube root of both the number and the variable component:

• For 27, since \( 27 = 3^3 \), the cube root is 3, as \( \sqrt[3]{27} = 3 \).
• For \( a^4 \), to find the cube root, we use the property that \( \sqrt[3]{a^4} = a^{4/3} \). This means finding a power such that when raised to 3, it results in \( a^4 \), yielding \( a^{4/3} \) as the solution.

By understanding these individual steps, we can confidently say the entire expression simplifies to \( 3a^{4/3} \). Breaking down radical expressions into their components makes it easier to see the path to their simplified forms.