Final answer:
The product of √5 (the square root of 5) and ³√5 (the cube root of 5) is found by expressing both roots as fractional exponents, then adding those exponents together, resulting in 5 raised to the five-sixths power, or 5^(5/6).
Step-by-step explanation:
The question asks to find the product of √5 (the square root of 5) and ³√5 (the cube root of 5). To do this, we'll need to express each root as an exponent. Recall that x² = √x, which means the square root of a number is the number to the 1/2 power. Similarly, the cube root of a number is the number to the 1/3 power.
To find the product of √5 * ³√5, we re-express the roots as fractional powers:
- √5 = 5^(1/2)
- ³√5 = 5^(1/3)
Multiplying these together, we use the property that when multiplying like bases, we add the exponents:
5^(1/2) * 5^(1/3) = 5^((1/2)+(1/3))
Adding the fractions, we get:
5^((3/6)+(2/6)) = 5^(5/6).
Therefore, the product of √5 and ³√5 is 5^(5/6), which is the number 5 raised to the five-sixths power.