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Chapter 7: Problem 86

Solve each equation. $$(a-1)^{1 / 3}=-3$$

Short Answer

Expert verified
The solution is \( a = -26 \).

Step by step solution

01

Isolate the term with the exponent

The given equation is \( (a-1)^{1/3} = -3 \). The term with the exponent is already isolated, so no further steps are needed for isolation.
02

Eliminate the exponent

To eliminate the cube root, raise both sides of the equation to the power of 3: \( ((a-1)^{1/3})^3 = (-3)^3 \). This simplifies to \( a-1 = -27 \).
03

Solve for the variable

Add 1 to both sides of the equation to isolate \( a \): \( a-1+1 = -27+1 \). This simplifies to \( a = -26 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Term
When solving an equation, the first step is to isolate the term with the variable you are solving for. Isolating the term means to get the specific term with the variable by itself on one side of the equation. In the case of the equation \( (a-1)^{1/3} = -3 \), the term \( (a-1)^{1/3} \) is already isolated. It is already by itself on the left-hand side of the equation. At this stage, we don't need to do any extra steps to isolate it since no other terms or coefficients are attached to the expression.
Eliminate the Exponent
The next step is to eliminate the exponent. Here, the variable is within a cube root, \( (a-1)^{1/3} \). To remove the cube root, we need to do the opposite operation, which is raising both sides to the power of 3. This action will cancel out the cube root:
\[((a-1)^{1/3})^3 = (-3)^3 \]
Simplifying both sides, we get:
\[(a-1) = -27 \]. By raising to the power of 3, the exponent on the left side is eliminated, leaving us with a linear equation.
Solve for the Variable
After eliminating the exponent, the equation becomes linear and simpler to solve. We now have \( a-1 = -27 \). The next step is to isolate the variable 'a'. To do this, we need to get rid of the -1 that is subtracted from 'a'. We can achieve this by adding 1 to both sides of the equation:
\[(a-1) + 1 = -27 + 1 \]
Simplifying both sides, we obtain:
\ a = -26 \.
This means the value of 'a' that satisfies the original equation is -26. Hence, the variable has been successfully isolated and solved.

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