Evaluate cube root of 2* fifth root of 2^3

Solution:

Step 1:

Compute the value of $2^3$. $\sqrt[3]{2} \cdot \sqrt[5]{2^3}$

Step 2:

Combine the radicals $\sqrt[3]{2} \cdot \sqrt[5]{2^3}$.

Step 2.1:

Convert to a common radical index of $15$.

Step 2.1.1:

Express $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$. $2^{\frac{1}{3}} \cdot \sqrt[5]{2^3}$

Step 2.1.2:

Change $2^{\frac{1}{3}}$ to $2^{\frac{5}{15}}$. $2^{\frac{5}{15}} \cdot \sqrt[5]{2^3}$

Step 2.1.3:

Represent $2^{\frac{5}{15}}$ as $\sqrt[15]{2^5}$. $\sqrt[15]{2^5} \cdot \sqrt[5]{2^3}$

Step 2.1.4:

Transform $\sqrt[5]{2^3}$ to $2^3^{\frac{1}{5}}$. $\sqrt[15]{2^5} \cdot 2^3^{\frac{1}{5}}$

Step 2.1.5:

Convert $2^3^{\frac{1}{5}}$ to $2^3^{\frac{3}{15}}$. $\sqrt[15]{2^5} \cdot 2^3^{\frac{3}{15}}$

Step 2.1.6:

Rewrite $2^3^{\frac{3}{15}}$ as $\sqrt[15]{2^3^3}$. $\sqrt[15]{2^5} \cdot \sqrt[15]{2^3^3}$

Step 2.2:

Apply the radical product rule. $\sqrt[15]{2^5 \cdot 2^3^3}$

Step 2.3:

Express $2^3$ as $2^3$. $\sqrt[15]{2^5 \cdot (2^3)^3}$

Step 2.4:

Calculate the exponentiation in $(2^3)^3$.

Step 2.4.1:

Use the exponentiation rule, $(a^m)^n = a^{mn}$. $\sqrt[15]{2^5 \cdot 2^{3 \cdot 3}}$

Step 2.4.2:

Multiply $3$ by $3$. $\sqrt[15]{2^5 \cdot 2^9}$

Step 2.5:

Combine exponents using the rule $a^m a^n = a^{m + n}$. $\sqrt[15]{2^{5 + 9}}$

Step 2.6:

Add the exponents $5$ and $9$. $\sqrt[15]{2^{14}}$

Step 3:

Calculate $2^{14}$. $\sqrt[15]{16384}$

Step 4:

Present the result in various forms.

Exact Form: $\sqrt[15]{16384}$ Decimal Form: $1.90968320\ldots$

Knowledge Notes:

The problem involves simplifying an expression with different roots, specifically a cube root and a fifth root. The process involves the following knowledge points:

1. Exponentiation: This is the process of raising a number to a power. In this problem, we raise $2$ to the power of $3$ to simplify the fifth root of $2^3$.

2. Radicals: A radical expression involves roots, such as square roots, cube roots, etc. The cube root of $2$ and the fifth root of $2^3$ are examples of radicals.

3. Rational Exponents: These are exponents that are fractions. The cube root of $2$ can be expressed as $2^{\frac{1}{3}}$ and the fifth root of $2^3$ as $2^3^{\frac{1}{5}}$.

4. Least Common Index: To combine radicals, we find a common index, which is the least common multiple of the indices of the given radicals. In this case, the least common index is $15$.

5. Product Rule for Radicals: This rule states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$. We use this rule to combine the two radicals into one.

6. Power Rule: The power rule, $(a^m)^n = a^{mn}$, is used to simplify expressions where an exponent is raised to another power.

7. Combining Exponents: When multiplying like bases, we add the exponents, as in $a^m \cdot a^n = a^{m+n}$.

8. Decimal Approximation: The exact form of a radical can be approximated to a decimal value for practical use, though it may be an irrational number.

Understanding these concepts is essential to simplify radical expressions and to perform operations with exponents and roots.