Multiply: [tex]$\sqrt[4]{x} \cdot \sqrt[4]{y^3}$[/tex]
Rewrite the expression using rational exponents with a common denominator.
[tex]$x^{-6} \cdot y^{\frac{3}{4}}$[/tex]
[tex]$x^{\frac{2}{12}} \cdot y^{\frac{9}{12}}$[/tex]
[tex]$x y^{\frac{11}{12}}$[/tex]
Answer (2)
The expression 4 x ⋅ 4 y 3 can be rewritten as x 4 1 ⋅ y 4 3 . Additionally, x 12 2 ⋅ y 12 9 simplifies to x 6 1 ⋅ y 4 3 . The other expressions remain unchanged as they are already in rational exponent form.
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Rewrite the first expression using rational exponents: 4 x ⋅ 4 y 3 = x 4 1 y 4 3 .
The second expression x − 6 ⋅ y 4 3 remains unchanged.
Simplify the third expression: x 12 2 ⋅ y 12 9 = x 6 1 y 4 3 .
The fourth expression x y 12 11 remains unchanged.
The simplified expressions are: x 4 1 y 4 3 , x − 6 y 4 3 , x 6 1 y 4 3 , x y 12 11 .
Explanation
Problem Analysis We are given several expressions involving exponents and radicals, and we need to simplify or rewrite them as requested. Let's address each expression one by one.
Rewriting with Rational Exponents First, we have the expression 4 x \t ⋅ 4 y 3 . We can rewrite this using rational exponents. Recall that n a = a n 1 . Therefore, we have 4 x = x 4 1 4 y 3 = y 4 3 So, 4 x ⋅ 4 y 3 = x 4 1 ⋅ y 4 3 .
Analyzing the Second Expression Next, we are given the expression x − 6 ⋅ y 4 3 . This expression is already written with rational exponents. There's nothing to simplify here unless we want to rewrite x − 6 as x 6 1 , but the problem doesn't ask for that.
Simplifying the Third Expression Now, consider the expression x 12 2 ⋅ y 12 9 . We can simplify the exponents by reducing the fractions: 12 2 = 6 1 12 9 = 4 3 So, the expression becomes x 6 1 ⋅ y 4 3 .
Analyzing the Fourth Expression Finally, we have the expression x y 12 11 . This expression is already in a simplified form. There's nothing to simplify here.
Final Results In summary:
4 x ⋅ 4 y 3 = x 4 1 y 4 3
x − 6 ⋅ y 4 3 remains as is.
x 12 2 ⋅ y 12 9 = x 6 1 y 4 3
x y 12 11 remains as is.
Examples
Understanding and manipulating exponents and radicals is crucial in many scientific and engineering fields. For instance, in physics, you might use these concepts to describe the relationship between the size of an object and its volume or surface area. If you're designing a spherical tank, knowing how the volume scales with the radius (which involves exponents) helps you determine how much material you need. Similarly, in chemistry, understanding exponential relationships is essential for studying reaction rates and radioactive decay.
