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Chapter 10: Problem 25

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

Short Answer

Expert verified
z = 16

Step by step solution

01

- Isolate the Radical

First, isolate the radical expression by subtracting 8 from both sides of the equation: \[ z^{1 / 4} + 8 = 10 \] Subtracting 8 from both sides: \[ z^{1 / 4} = 2 \]
02

- Eliminate the Radical

Raise both sides of the equation to the power of 4 in order to eliminate the 1/4 exponent: \[ (z^{1 / 4})^4 = 2^4 \] Simplifying, we get: \[ z = 16 \]

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Key Concepts

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

Isolating the Variable
To solve the equation involving exponents, the first step is isolating the variable with the radical. In the exercise, the equation is given as:
\(z^{1 / 4} + 8 = 10\)
  • Subtracting 8 from both sides simplifies the equation and isolates the variable within the radical expression.
  • The updated equation becomes: \(z^{1 / 4} = 2\)
By isolating the variable, we can directly address the term that contains the exponent. This simplification makes it easier to apply further mathematical operations.
Radical Expressions
Radical expressions are expressions that include roots, such as square roots or fourth roots. In our example, the term \(z^{1/4}\) is a fourth root expression.
To handle these expressions, you often need to clear the root by using exponents. In our exercise, once we isolate the radical expression: \(z^{1/4} = 2\), we then proceed to eliminate the radical by raising both sides of the equation to the power of 4:
  • Raising \(z^{1/4}\) to the power of 4 and \(2\) to the power of 4: \((z^{1/4})^4 = 2^4\)
  • This simplifies to \(z = 16\)
This process effectively removes the root, leaving you with a simple equation that is much easier to solve.
Exponent Rules
Understanding exponent rules is crucial when solving equations with radical expressions. In this example, we use one of the basic exponent rules: \((a^{m/n})^n = a^m\)
Breaking it down:
  • We need to deal with \(z^{1/4}\). To eliminate the fraction in the exponent, raise both sides to the power of 4.
  • This can be written mathematically as: \((z^{1/4})^4 = 2^4\)
  • Using the exponent rule: \((a^{m/n})^n = a^m\), simplifies to \z = 16.
By applying these rules correctly, you convert complex radical expressions into simpler forms, allowing for straightforward solutions to equations.

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