Answer:
Explanation:
You want to identify the solutions to the equations shown:
- √(x²+7) = 4
- ∛(1 -x) = -1
- ∜(x-2) = 2
(a) Square root
In each case, the first step is to eliminate the radical by raising both sides of the equation to the appropriate power.
x² +7 = 4² . . . . . . square both sides
x² = 16 -7 = 9 . . . . subtract 7
x = ±3 . . . . . . . . . . . take the square root
A suitable value of x is -3.
(b) Cube root
Cube both sides of the equation:
1 -x = -1
2 = x . . . . . . . add x+1 to both sides
A suitable value of x is 2.
(c) Fourth root
Raise both sides of the equation to the 4th power:
x -2 = 2⁴
x = 16 +2 = 18 . . . . . add 2 to both sides
A suitable value of x is 18.
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Additional comment
As a rule, equations are solved by reversing the operations done to the variable. The inverse operation of taking a root is raising to a power.
It can be helpful to understand a root as a fractional power. You "clear fractions" by multiplying by the denominator. An exponent is multiplied by raising the expression to a power: (a^b)^c = a^(bc).
For fractional powers, such as 4th roots, this looks like ...
(a^(1/4))^4 = a^(4/4) = a
Raising a 4th root to the 4th power gives the original number.