Simplify (2^4*6^4)^(-1/4)

Solution:

Step 1:

Compute $2^4$. The expression becomes $\left(16 \cdot 6^4\right)^{-\frac{1}{4}}$.

Step 2:

Compute $6^4$. The expression becomes $\left(16 \cdot 1296\right)^{-\frac{1}{4}}$.

Step 3:

Calculate the product of $16$ and $1296$. The expression becomes $\left(20736\right)^{-\frac{1}{4}}$.

Step 4:

Apply the negative exponent rule, which states $b^{-n} = \frac{1}{b^n}$. The expression becomes $\frac{1}{\left(20736\right)^{\frac{1}{4}}}$.

Step 5:

Simplify the expression under the radical.

Step 5.1:

Express $20736$ as a power of $12$, which is $\left(12\right)^4$. The expression becomes $\frac{1}{\left(\left(12\right)^4\right)^{\frac{1}{4}}}$.

Step 5.2:

Use the power of a power rule, which states $\left(a^m\right)^n = a^{mn}$. The expression becomes $\frac{1}{\left(12\right)^{4 \cdot \frac{1}{4}}}$.

Step 5.3:

Reduce the exponent by canceling out the common factor of $4$.

Step 5.3.1:

Perform the cancellation. The expression becomes $\frac{1}{\left(12\right)^{\cancel{4} \cdot \frac{1}{\cancel{4}}}}$.

Step 5.3.2:

Simplify the expression to $\frac{1}{\left(12\right)^1}$.

Step 5.4:

Evaluate the final exponent. The simplified expression is $\frac{1}{12}$.

Step 6:

Present the final result in different forms. The exact form is $\frac{1}{12}$, and the decimal form is approximately $0.0833$.

Knowledge Notes:

1. Exponentiation: Raising a number to a power multiplies the base by itself a specified number of times. For example, $2^4$ means $2 \times 2 \times 2 \times 2$.

2. Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, $b^{-n} = \frac{1}{b^n}$.

3. Power of a Power Rule: When raising a power to another power, multiply the exponents. This is denoted as $\left(a^m\right)^n = a^{mn}$.

4. Simplification: The process of reducing an expression to its simplest form by performing all possible operations and canceling common factors.

5. Radicals and Rational Exponents: A radical can be expressed as a rational exponent, where the nth root of a number is the same as raising that number to the power of $\frac{1}{n}$. For example, $\sqrt[4]{20736}$ is equivalent to $\left(20736\right)^{\frac{1}{4}}$.

6. Multiplication and Division of Powers: When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.

7. Simplifying Expressions: To simplify an expression involving exponents, it is often useful to rewrite numbers as powers of a common base and then apply the rules of exponents to simplify.