Simplify ( fourth root of 4)( cube root of 4)

Solution:

Step 1:

Express the number 4 as $2^2$. Thus, the expression becomes $\sqrt[4]{2^2} \cdot \sqrt[3]{4}$.

Step 2:

Transform $\sqrt[4]{2^2}$ into $\sqrt{\sqrt{2^2}}$. The expression now reads $\sqrt{\sqrt{2^2}} \cdot \sqrt[3]{4}$.

Step 3:

Extract terms from under the radical sign, assuming all numbers are positive real numbers. The expression simplifies to $\sqrt{2} \cdot \sqrt[3]{4}$.

Step 4:

Proceed to multiply $\sqrt{2} \cdot \sqrt[3]{4}$.

Step 4.1:

Convert the expression to have a common index of 6.

Step 4.1.1:

Using the rule $\sqrt[n]{a^x} = a^{\frac{x}{n}}$, rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$. The expression is now $2^{\frac{1}{2}} \cdot \sqrt[3]{4}$.

Step 4.1.2:

Express $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$. The expression updates to $2^{\frac{3}{6}} \cdot \sqrt[3]{4}$.

Step 4.1.3:

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^3}$. The expression becomes $\sqrt[6]{2^3} \cdot \sqrt[3]{4}$.

Step 4.1.4:

Use the rule $\sqrt[n]{a^x} = a^{\frac{x}{n}}$ to convert $\sqrt[3]{4}$ into $4^{\frac{1}{3}}$. The expression is now $\sqrt[6]{2^3} \cdot 4^{\frac{1}{3}}$.

Step 4.1.5:

Change $4^{\frac{1}{3}}$ to $4^{\frac{2}{6}}$. The expression updates to $\sqrt[6]{2^3} \cdot 4^{\frac{2}{6}}$.

Step 4.1.6:

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^2}$. The expression is now $\sqrt[6]{2^3} \cdot \sqrt[6]{4^2}$.

Step 4.2:

Combine the terms using the product rule for radicals to get $\sqrt[6]{2^3 \cdot 4^2}$.

Step 4.3:

Represent the number 4 as $2^2$. The expression becomes $\sqrt[6]{2^3 \cdot (2^2)^2}$.

Step 4.4:

Apply the power rule to the exponents in $(2^2)^2$.

Step 4.4.1:

Using the power rule $(a^m)^n = a^{mn}$, the expression simplifies to $\sqrt[6]{2^3 \cdot 2^{2 \cdot 2}}$.

Step 4.4.2:

Multiply 2 by 2 to get $\sqrt[6]{2^3 \cdot 2^4}$.

Step 4.5:

Combine the exponents using the power rule $a^m \cdot a^n = a^{m+n}$, resulting in $\sqrt[6]{2^{3+4}}$.

Step 4.6:

Add the exponents 3 and 4 to obtain $\sqrt[6]{2^7}$.

Step 5:

Raise 2 to the power of 7 to get $\sqrt[6]{128}$.

Step 6:

Decompose 128 into $2^6 \cdot 2$.

Step 6.1:

Factor out 64 from 128, leading to $\sqrt[6]{64 \cdot 2}$.

Step 6.2:

Express 64 as $2^6$, resulting in $\sqrt[6]{2^6 \cdot 2}$.

Step 7:

Extract terms from under the radical to get $2 \cdot \sqrt[6]{2}$.

Step 8:

Present the result in various forms:

• Exact Form: $2 \cdot \sqrt[6]{2}$
• Decimal Form: Approximately $2.24492409 \ldots$

Knowledge Notes:

• Radicals and Exponents: The radical expression $\sqrt[n]{a}$ is equivalent to $a^{\frac{1}{n}}$. When dealing with radicals, it's often useful to express numbers in their prime factorized form to simplify the radical.

• Simplifying Radicals: To simplify a radical, one can pull out squares, cubes, etc., from under the radical sign. If the index of the radical and the exponent have a common multiple, it is possible to rewrite them with a common index to further simplify.

• Product Rule for Radicals: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$ allows us to combine radicals with the same index.

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

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

• Exact vs. Decimal Form: The exact form of a number is the simplified radical form, while the decimal form is an approximate value obtained by calculating the radical expression to a certain number of decimal places.