Question

Simplify each expression. $$\left(27 x^{6}\right)^{2 / 3}$$

   Simplify each expression.
$$\left(27 x^{6}\right)^{2 / 3}$$

College Algebra

College Algebra

R. David Gustafson,… 11th Edition

Chapter 0, Problem 110

Instant Answer

Solved by Expert Casey Evans

07/17/2020

Step 1

Step 1: We can rewrite the expression as two separate terms raised to the power of 2/3: $$\left(27 x^{6}\right)^{2 / 3} = 27^{2/3} \cdot x^{6(2/3)}$$ Show more…

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Simplify each expression. $$\left(27 x^{6}\right)^{2 / 3}$$

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Key Concepts

Fractional Exponents

Fractional exponents represent both an exponentiation and a radical operation. An exponent in the form of m/n indicates that you should take the nth root of the base and then raise it to the mth power, or vice versa. This interpretation helps in simplifying expressions where both radicals and powers are present.

Laws of Exponents

The laws of exponents, including the power of a product and the power of a power rules, assist in breaking down and simplifying expressions that involve exponents. These rules allow you to distribute an exponent over a product or combine exponents when raising an exponential expression to another power.

Radical-Exponent Equivalence

Radical-exponent equivalence is the principle that ties radicals to fractional exponents, such that taking the nth root of a number is the same as raising it to the power of 1/n. This equivalence is essential for converting between radical notation and exponent notation, thereby simplifying calculations and manipulations of algebraic expressions.

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