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Chapter 7: Problem 23

Simplify. $$ \sqrt[4]{z^{8}} $$

Short Answer

Expert verified
The simplified form is \( z^2 \).

Step by step solution

01

Understand the Expression

The expression given is \( \sqrt[4]{z^8} \). This is a fourth root of \( z^8 \). The problem asks us to simplify this expression.
02

Apply the Root-Power Rule

According to the rules of exponents, \( \sqrt[n]{a^m} = a^{m/n} \). In this case, we apply it as: \( \sqrt[4]{z^8} = z^{8/4} \).
03

Simplify the Fraction

Simplify the fraction \( 8/4 \). The result of \( 8 \div 4 \) is \( 2 \). Thus, \( z^{8/4} = z^2 \).
04

Write the Final Simplified Expression

The simplified form of the expression \( \sqrt[4]{z^8} \) is \( z^2 \).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is like tidying up a room. You're organizing and reducing the math to its simplest form. This makes calculations easier and expressions more understandable. In the exercise, you started with a complex mathematical expression: \( \sqrt[4]{z^8} \). The end goal is to simplify it to make it as elementary as possible. Here's how:
  • Identify parts of the expression that can be simplified.
  • Use mathematical rules, such as those dealing with roots and powers, to reduce the expression.
  • Simplify any arithmetic, like fractions, to arrive at a final, streamlined expression.
By doing this, you turn a possibly daunting problem into something far more manageable. Just like organizing a cluttered space, simplifying expressions involves recognizing patterns and applying logical steps to achieve neatness.
Root-Power Rule
The Root-Power Rule is a powerful tool that helps simplify expressions involving roots and exponents. The rule states: \( \sqrt[n]{a^m} = a^{m/n} \). This rule can drastically transform complex expressions into straightforward powers. In the original problem, you applied the rule to \( \sqrt[4]{z^8} \). You expressed it as \( z^{8/4} \). This step changes a root scenario into a simpler power form, which is easier to work with. Benefits of the Root-Power Rule:
  • Converts roots to fractional exponents, simplifying high-root calculations.
  • Allows straightforward simplification of fractions.
  • Makes further operations, such as multiplication or division of powers, much simpler.
This method effortlessly breaks down complex roots, making it a go-to strategy for any root-related simplification task.
Fraction Simplification
Fraction simplification helps make expressions less cumbersome by reducing a fraction to its simplest form. It is like simplifying a recipe 鈥� removing any excess without altering the outcome. In the exercise, you simplified the fraction \( 8/4 \), which is part of the expression \( z^{8/4} \). Steps to simplify a fraction:
  • Find the greatest common divisor (GCD) for the numerator and the denominator.
  • Divide both by their GCD to reduce the fraction.
For \( 8/4 \):
  • The GCD of 8 and 4 is 4.
  • Dividing both terms by 4 gives \( 2/1 \) or simply \( 2 \).
The expression thus simplifies to \( z^2 \). Fraction simplification is vital, not just for neatness but for simplifying further math operations. It turns complex expressions into the simplest version, ready for evaluation or further manipulation.

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