Simplify (7 cube root of 3)/( cube root of 9)

Solution:

Step 1:

Merge the cube roots into a single radical expression: $7 \sqrt[3]{\frac{3}{9}}$.

Step 2:

Simplify the fraction under the radical by identifying common factors.

Step 2.1:

Express the numerator as a product involving the common factor: $7 \sqrt[3]{\frac{3 \cdot 1}{9}}$.

Step 2.2:

Proceed to eliminate the common factors.

Step 2.2.1:

Factor out the common factor from the denominator: $7 \sqrt[3]{\frac{3 \cdot 1}{3 \cdot 3}}$.

Step 2.2.2:

Cross out the common factors: $7 \sqrt[3]{\frac{\cancel{3} \cdot 1}{\cancel{3} \cdot 3}}$.

Step 2.2.3:

Rewrite the simplified expression: $7 \sqrt[3]{\frac{1}{3}}$.

Step 3:

Express the cube root of a fraction as a fraction of cube roots: $7 \frac{\sqrt[3]{1}}{\sqrt[3]{3}}$.

Step 4:

Recognize that the cube root of $1$ is $1$: $7 \frac{1}{\sqrt[3]{3}}$.

Step 5:

Rationalize the denominator by multiplying by the conjugate: $7 \left( \frac{1}{\sqrt[3]{3}} \cdot \frac{\left(\sqrt[3]{3}\right)^{2}}{\left(\sqrt[3]{3}\right)^{2}} \right)$.

Step 6:

Simplify the expression by combining terms in the denominator.

Step 6.1:

Multiply the numerator and denominator by the square of the cube root: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{\sqrt[3]{3} \cdot \left(\sqrt[3]{3}\right)^{2}}$.

Step 6.2:

Express the first term in the denominator with an exponent: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{\left(\sqrt[3]{3}\right)^{1} \cdot \left(\sqrt[3]{3}\right)^{2}}$.

Step 6.3:

Apply the exponent rule for multiplication: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{\left(\sqrt[3]{3}\right)^{1+2}}$.

Step 6.4:

Add the exponents: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{\left(\sqrt[3]{3}\right)^{3}}$.

Step 6.5:

Convert the cube root to an exponent and simplify.

Step 6.5.1:

Rewrite the cube root as an exponent: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{\left(3^{\frac{1}{3}}\right)^{3}}$.

Step 6.5.2:

Apply the power of a power rule: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{3^{\frac{1}{3} \cdot 3}}$.

Step 6.5.3:

Simplify the exponent: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{3^{\frac{3}{3}}}$.

Step 6.5.4:

Cancel out the common factor in the exponent.

Step 6.5.4.1:

Eliminate the common factor: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{3^{\frac{\cancel{3}}{\cancel{3}}}}$.

Step 6.5.4.2:

Rewrite the simplified expression: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{3^{1}}$.

Step 6.5.5:

Evaluate the exponent: $7 \frac{\left(\sqrt[3]{3}\right)^{2}}{3}$.

Step 7:

Simplify the numerator by expressing the square of the cube root as a cube root of a square.

Step 7.1:

Rewrite the square of the cube root: $7 \frac{\sqrt[3]{3^{2}}}{3}$.

Step 7.2:

Calculate the square of $3$: $7 \frac{\sqrt[3]{9}}{3}$.

Step 8:

Combine the constant with the simplified radical: $\frac{7 \sqrt[3]{9}}{3}$.

Step 9:

Present the final result in various forms.

Exact Form: $\frac{7 \sqrt[3]{9}}{3}$

Decimal Form: $4.85352892 \ldots$

Knowledge Notes:

The problem involves simplifying a fraction that contains cube roots. The key knowledge points and steps in this process include:

1. Combining Radicals: When simplifying expressions with radicals, it's often useful to combine them into a single radical to make simplification easier.

2. Simplifying Fractions: Look for common factors in the numerator and denominator that can be canceled out to simplify the fraction.

3. Rationalizing the Denominator: When a radical is in the denominator, we often multiply by a form of one to eliminate the radical from the denominator. This process is known as rationalizing the denominator.

4. Exponent Rules: Several exponent rules are used in this process, including:

   • The power rule: $a^m \cdot a^n = a^{m+n}$.

   • The power of a power rule: $(a^m)^n = a^{m \cdot n}$.

   • The fact that any number raised to the power of one is itself: $a^1 = a$.

5. Cube Roots and Exponents: The cube root of a number can be expressed as that number raised to the $\frac{1}{3}$ power. Similarly, the cube root of a cube (e.g., $\sqrt[3]{3^3}$) is simply the number itself (in this case, $3$).

6. Simplifying Radical Expressions: When simplifying expressions involving radicals, it's often helpful to rewrite powers inside the radical to see if further simplification is possible.

By understanding and applying these principles, the original expression can be simplified to its most reduced form.