Simplify the Radical Expression ( cube root of 5)^3

Solution:

Step 1:

Convert the cube root into an exponent form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. Thus, $\sqrt[3]{5}$ becomes $5^{\frac{1}{3}}$. The expression now is $\left(5^{\frac{1}{3}}\right)^{3}$.

Step 2:

Use the exponent multiplication rule, which states $\left(a^{m}\right)^{n} = a^{m \cdot n}$. Apply this to get $5^{\frac{1}{3} \cdot 3}$.

Step 3:

Multiply the exponents to simplify the expression, resulting in $5^{\frac{3}{3}}$.

Step 4:

Simplify the exponent by canceling out like terms.

Step 4.1:

Remove the common factors to simplify the fraction in the exponent: $5^{\frac{\cancel{3}}{\cancel{3}}}$.

Step 4.2:

The simplified expression is now $5^{1}$.

Step 5:

Evaluate the expression with the simplified exponent to get the final answer: $5$.

Knowledge Notes:

To simplify radical expressions, one should be familiar with the following concepts:

1. Radical to Exponent Conversion: A radical expression can be converted to an exponent form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$, where $n$ is the index of the radical and $x$ is the exponent of the term inside the radical.

2. Exponent Rules:

   • Power Rule: When raising a power to a power, you multiply the exponents, as in $\left(a^{m}\right)^{n} = a^{m \cdot n}$.

   • Product of Powers: When multiplying like bases, you add the exponents, as in $a^{m} \cdot a^{n} = a^{m+n}$.

   • Quotient of Powers: When dividing like bases, you subtract the exponents, as in $\frac{a^{m}}{a^{n}} = a^{m-n}$.

   • Zero Exponent: Any non-zero base raised to the zero power is equal to one, as in $a^{0} = 1$.

   • Negative Exponent: A negative exponent indicates a reciprocal, as in $a^{-n} = \frac{1}{a^{n}}$.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and denominator can be canceled out.

4. Evaluating Exponents: Once the exponent is simplified, the expression can be evaluated by raising the base to the power of the exponent.