Complete the following exercise:
316⋅16=
To solve this problem, we will simplify the expression 316⋅16 using the rules for exponents and roots.
First, consider the inner square root 16. We know that:
16=161/2
Next, we address the cube root term 316.
Express 16 as 161/2, then:
- 316=3161/2
Convert to exponents:
3161/2=(161/2)1/3=16(1/2)⋅(1/3)=161/6
- The other term is 16=161/2
Convert to exponents:
161/2=(161/2)1/2=16(1/2)⋅(1/2)=161/4
Now, multiply these results:
161/6⋅161/4
Using the product rule for exponents (am⋅an=am+n), combine the exponents:
161/6+1/4
Find the common denominator to add the fractions:
- 1/6=2/12
- 1/4=3/12
1/6+1/4=2/12+3/12=5/12
Thus, the expression becomes:
165/12
Therefore, the simplified expression is 16125.
16125