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

The question is asking you to perform operations on exponential expressions and then simplify the result. You are given a complex-looking expression (2^4*6^4)^(-1/4), and the goal is to apply the power of a power rule, followed by the simplification of the expression that results from this operation. You are likely expected to express the simplified form without any negative exponents.

${((2^{4} \cdot 6^{4}))}^{- \frac{1}{4}}$

Answer

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Solution:

Step 1:

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

Step 2:

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

Step 3:

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

Step 4:

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

Step 5:

Simplify the expression under the radical.

Step 5.1:

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

Step 5.2:

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

Step 5.3:

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

Step 5.3.1:

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

Step 5.3.2:

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

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:

Exponentiation: Raising a number to a power multiplies the base by itself a specified number of times. For example, $2^{4}$ means $2 \times 2 \times 2 \times 2$ .
Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, $b^{- n} = \frac{1}{b^{n}}$ .
Power of a Power Rule: When raising a power to another power, multiply the exponents. This is denoted as ${(a^{m})}^{n} = a^{m n}$ .
Simplification: The process of reducing an expression to its simplest form by performing all possible operations and canceling common factors.
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 ${(20736)}^{\frac{1}{4}}$ .
Multiplication and Division of Powers: When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.
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.