The cube root of x times the square root of x can be rewritten as x to the
The Correct Answer and Explanation is:
The cube root of xxx times the square root of xxx can be rewritten as:x3⋅x=x13⋅x12=x56\sqrt[3]{x} \cdot \sqrt{x} = x^{\frac{1}{3}} \cdot x^{\frac{1}{2}} = x^{\frac{5}{6}}3x⋅x=x31⋅x21=x65
Textbook-Style Explanation
In algebra, radicals such as square roots and cube roots can be rewritten using rational exponents. This allows us to apply the laws of exponents more easily.
Let’s begin with the original expression:x3⋅x\sqrt[3]{x} \cdot \sqrt{x}3x⋅x
This expression involves two radicals: the cube root of xxx, and the square root of xxx. To simplify, we first rewrite both radicals using exponent notation.
- The cube root of xxx is written as x1/3x^{1/3}x1/3
- The square root of xxx is written as x1/2x^{1/2}x1/2
So the original expression becomes:x1/3⋅x1/2x^{1/3} \cdot x^{1/2}x1/3⋅x1/2
Now we apply the Product of Powers Property, which states that when multiplying two expressions with the same base, we can add the exponents:x1/3⋅x1/2=x1/3+1/2x^{1/3} \cdot x^{1/2} = x^{1/3 + 1/2}x1/3⋅x1/2=x1/3+1/2
To add the exponents, we need a common denominator. The least common denominator of 3 and 2 is 6, so we convert both fractions:
- 13=26\frac{1}{3} = \frac{2}{6}31=62
- 12=36\frac{1}{2} = \frac{3}{6}21=63
Now add:x2/6+3/6=x5/6x^{2/6 + 3/6} = x^{5/6}x2/6+3/6=x5/6
So the expression simplifies to:x5/6x^{5/6}x5/6
This simplified form is useful in algebraic manipulation and calculus because rational exponents are easier to differentiate or integrate than radical expressions. Converting between radical form and exponent form is a foundational skill in algebra that helps make more advanced math accessible.
