Simplify the expression: [tex]$\sqrt[4]{16 x^8 y^4} \div \sqrt[3]{8 x^6 y^3}$[/tex]
Answer (2)
Rewrite the expression using exponents.
Simplify the terms inside the parentheses.
Apply the power rule to both numerator and denominator.
Simplify the exponents and cancel out the common terms, resulting in 1 .
Explanation
Understanding the Problem We are given the expression 4 16 x 8 y 4 ÷ 3 8 x 6 y 3 . Our goal is to simplify this expression. We will assume that x and y are positive.
Rewriting with Exponents First, let's rewrite the expression using exponents: 3 8 x 6 y 3 4 16 x 8 y 4 = ( 8 x 6 y 3 ) 1/3 ( 16 x 8 y 4 ) 1/4 .
Simplifying the Terms Now, we simplify the terms inside the parentheses. We know that 16 = 2 4 and 8 = 2 3 . So, we can rewrite the expression as: ( 2 3 x 6 y 3 ) 1/3 ( 2 4 x 8 y 4 ) 1/4 .
Applying the Power Rule Next, we apply the power rule, which states that ( a m ) n = a mn . Applying this rule to both the numerator and the denominator, we get: 2 3 ( 1/3 ) x 6 ( 1/3 ) y 3 ( 1/3 ) 2 4 ( 1/4 ) x 8 ( 1/4 ) y 4 ( 1/4 ) .
Simplifying Exponents Now, we simplify the exponents: 2 1 x 2 y 1 2 1 x 2 y 1 = 2 x 2 y 2 x 2 y .
Canceling Common Terms Finally, we cancel out the common terms in the numerator and the denominator: 2 x 2 y 2 x 2 y = 1 .
Final Answer Therefore, the simplified expression is 1 .
Examples
Imagine you are calculating the volume of two different cubes. One cube's volume is expressed as 4 16 x 8 y 4 and the other as 3 8 x 6 y 3 . If you want to find the ratio of the first cube's volume to the second cube's volume, you would perform the division 4 16 x 8 y 4 ÷ 3 8 x 6 y 3 . Simplifying this expression helps you easily determine the ratio, which in this case is 1, meaning the volumes are equal.
The expression 4 16 x 8 y 4 ÷ 3 8 x 6 y 3 simplifies to 1 after applying exponent rules and canceling common terms.
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