a To find the cube root of 27, we must consider the value that, when multiplied by itself 3 times, equals 27.
sqrt(27)=x ⇔ 27=x^3
To help us find this value of x, we can list the cubes of the integers from 1 to 5.
x
x^3
1
1
2
8
3
27
4
64
5
125
From the table, we can see that 3^3=27. Therefore, sqrt(27)=3.
b Similar to Part A, we can write the given expression with an exponent.
sqrt(144)=x ⇔ 144=x^2
We can use the guess-and-check method to find the value of x that satisfies this equation. We need a number that, when multiplied by itself 2 times, equals 144. Let's start with x=10 because our number is greater than 100.
10^2 = 10* 10= 100
Now let's try with x=11 and x=12.
11^2 = 11*11 = 121
12^2=12*12=144
Therefore, sqrt(144)=12.
c Similar to previous parts, we can rewrite the given expression as an equation.
sqrt(3^2)=x ⇔ 3^2=x^2
As we can see to get true statement x must equal 3, so sqrt(3^2)=3.
d One last time, let's rewrite the given expression.
sqrt(2^4)=x ⇔ 2^4=x^4
Therefore x=2, so the expression is equal to 2.
