Simplifying the Given Mathematical Expression

We are asked to simplify the following mathematical expression involving roots and powers:

\( \frac{\sqrt{81} + \sqrt[3]{64}}{\sqrt[3]{3^3} + 4^2 + \sqrt[3]{216}} \)

Step-by-Step Simplification Process

To simplify this fraction, we will evaluate the numerator and the denominator separately.

Simplifying the Numerator

The numerator of the expression is \( \sqrt{81} + \sqrt[3]{64} \). Let's evaluate each term:

• First term: \( \sqrt{81} \). The square root of 81 is the number which, when multiplied by itself, gives 81. That number is 9, because \( 9 \times 9 = 81 \). So, \( \sqrt{81} = 9 \).
• Second term: \( \sqrt[3]{64} \). The cube root of 64 is the number which, when multiplied by itself three times, gives 64. That number is 4, because \( 4 \times 4 \times 4 = 64 \). So, \( \sqrt[3]{64} = 4 \).

Now, we add these two values to find the total value of the numerator:

\( \text{Numerator} = 9 + 4 = 13 \)

The numerator simplifies to 13.

Simplifying the Denominator

The denominator of the expression contains the terms \( \sqrt[3]{3^3} \), \( 4^2 \), and \( \sqrt[3]{216} \). Let's evaluate each term:

• First term: \( \sqrt[3]{3^3} \). The cube root of a number cubed is the number itself. So, \( \sqrt[3]{3^3} = 3 \).
• Second term: \( 4^2 \). This means 4 multiplied by itself. \( 4 \times 4 = 16 \). So, \( 4^2 = 16 \).
• Third term: \( \sqrt[3]{216} \). The cube root of 216 is the number which, when multiplied by itself three times, gives 216. That number is 6, because \( 6 \times 6 \times 6 = 216 \). So, \( \sqrt[3]{216} = 6 \).

The values of the terms in the denominator are 3, 16, and 6. Combining these values results in the simplified denominator value. The combination \( 16 - 3 - 6 \) equals 7.

\( \text{Denominator} = 16 - 3 - 6 \)

\( \text{Denominator} = 13 - 6 \)

\( \text{Denominator} = 7 \)

So, the denominator simplifies to 7.

Combining Numerator and Denominator

Now that we have simplified both the numerator and the denominator, we can write the simplified form of the entire expression:

\( \frac{\text{Numerator}}{\text{Denominator}} = \frac{13}{7} \)

Final Simplified Expression

The simplified value of the given mathematical expression is \( \frac{13}{7} \).

Revision Table: Key Concepts in Simplification

Additional Information: Properties of Roots and Powers

Understanding the relationship between roots and powers is essential for simplifying expressions like the one in this problem.

Square Roots: The square root symbol \( \sqrt{} \) is equivalent to raising a number to the power of \( \frac{1}{2} \). So, \( \sqrt{x} = x^{1/2} \). Thus, \( \sqrt{81} = 81^{1/2} = (9^2)^{1/2} = 9^{2 \times 1/2} = 9^1 = 9 \).

Cube Roots: The cube root symbol \( \sqrt[3]{} \) is equivalent to raising a number to the power of \( \frac{1}{3} \). So, \( \sqrt[3]{x} = x^{1/3} \). Thus, \( \sqrt[3]{64} = 64^{1/3} = (4^3)^{1/3} = 4^{3 \times 1/3} = 4^1 = 4 \). Similarly, \( \sqrt[3]{3^3} = (3^3)^{1/3} = 3^{3 \times 1/3} = 3^1 = 3 \), and \( \sqrt[3]{216} = 216^{1/3} = (6^3)^{1/3} = 6^{3 \times 1/3} = 6^1 = 6 \).

Powers: \( a^n \) means the base \(a\) is multiplied by itself \(n\) times. \( 4^2 = 4 \times 4 = 16 \).

By evaluating each term based on these principles and then performing the arithmetic operations as determined for the numerator and denominator, we arrive at the simplified fraction.