Question

Which expression is equivalent to rootindex 4 startroot startfraction 16 x superscript 11 baseline y superscript 8 baseline over 81 x superscript 7 baseline y superscript 6 baseline endfraction endroot? assume x greater-than 0 and y not-equals 0.

Answer 1

The expression equivalent to the fourth root of √¹⁶ⁱ¹¹⁹⁸/81x⁷y⁶ is 2x³/9y³. This is done by taking the fourth root of the numbers and subtracting the exponents for the variables.

Step-by-step explanation:

The student is asking how to simplify the fourth root of a quotient where the bases are variables raised to a power. The expression given is the fourth root of √¹⁶ⁱ¹¹⁹⁸ over 81x⁷y⁶. By applying the property that the fourth root of a number is the same as raising that number to the power of 0.25, and using the rules for exponents, we can simplify the expression.

First, the fourth root can also be calculated by taking two square roots in succession. As for the variables inside the root, we know that when dividing like bases, we subtract the exponents (e.g., xⁱ² / x = xⁱ¹). Appyling fraction and exponent rules, the expression can be simplified to:

• √⁴(√⁴(¹⁶/₈₁)) × √⁴x⁴ y²

In this expression, √¹⁶ is 2, ₈₁ is 9, and we have the powers of x and y that have been adjusted by subtracting the exponents.

After simplification, the result is ¹x³ × ³y³, or simply 2x³/9y³.