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Chapter 7: Problem 27

Solve. $$ z^{1 / 4}+8=10 $$

Short Answer

Expert verified
The solution is \( z = 16 \).

Step by step solution

01

Isolate the term with the variable

Start by isolating the term that contains the variable. Subtract 8 from both sides of the equation: \[ z^{1 / 4} + 8 - 8 = 10 - 8 \] This simplifies to: \[ z^{1 / 4} = 2 \]
02

Eliminate the exponent

Raise both sides of the equation to the 4th power to eliminate the fractional exponent: \[ \left( z^{1 / 4} \right)^4 = 2^4 \] Simplify both sides: \[ z = 16 \]
03

Verify the solution

To ensure the solution is correct, substitute \( z = 16 \) back into the original equation: \[ (16)^{1 / 4} + 8 = 10 \] Since \( (16)^{1 / 4} = 2 \), the equation becomes: \[ 2 + 8 = 10 \] This is true, so the solution is correct.

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Key Concepts

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

isolating the variable
When solving equations, one of the most important steps is to isolate the variable. This means you want to get the variable by itself on one side of the equation. For our equation, we start with: \[ z^{1 / 4} + 8 = 10 \]. To isolate the variable term \( z^{1 / 4} \), we need to remove the constant term on the same side. We do this by performing the inverse operation. Here, the constant term is '+8'. Therefore, we subtract 8 from both sides of the equation:
\[ z^{1 / 4} + 8 - 8 = 10 - 8 \]. This simplifies to: \[ z^{1 / 4} = 2 \]. Now, the variable term is isolated, and we can move on to eliminate the exponent.
fractional exponents
Fractional exponents are a way to express roots. For example, \( a^{1/n} \) means the n-th root of \( a \). In our original equation, the term \( z^{1 / 4} \) represents the fourth root of \( z \). To solve for \( z \), we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the power that is the inverse of the fractional exponent. Since our exponent is \( 1 / 4 \), we use \( 4 \) as the power: \[ \left( z^{1 / 4} \right)^4 = 2^4 \]. Raising both sides to the 4th power simplifies to: \[ z = 16 \]. This operation effectively 鈥渃ancels out鈥� the fractional exponent, leaving \( z \) by itself.
verifying solutions
After solving an equation, it is crucial to verify that the solution is correct. Verifying a solution ensures that no mistakes were made during the solving process. For our solved value \( z = 16 \), we substitute it back into the original equation to check our work: \[ (16)^{1 / 4} + 8 = 10 \]. We know that the fourth root of \( 16 \) is \( 2 \), because: \[ (16)^{1 / 4} = 2 \]. Substituting this value back, the equation becomes: \[ 2 + 8 = 10 \]. Since the left-hand side equals the right-hand side, our solution is verified as correct.

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Most popular questions from this chapter

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