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Chapter 9: Problem 32

Simplify the radical expression. Use absolute value signs, if appropriate. $$\sqrt{49 z^{5}}$$

Short Answer

Expert verified
The simplified expression is \( 7|z^{2}| \times \sqrt{z} \)

Step by step solution

01

Factor 49

The number 49 is a perfect square, and its square root is 7. So we can write the square root of 49 as 7.
02

Simplify the variable under the square root

The variable in the expression is \( z^5 \). We can rewrite this as \( z^2 \times z^2 \times z \). So it simplifies to \( z^2 \times z \).
03

Take the square root

The square root of \( z^2 \) is \( |z| \) (the absolute value of z). The square root of \( z \) can stay under the radical sign. Thus, the simplified form of the expression will be \( 7|z^{2}| \times \sqrt{z} \). Here, we use absolute value signs for z because the square root of squared value is always positive.

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Key Concepts

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

Understanding Absolute Value
Absolute value refers to the "non-negative" measure of a real number. It is simply how far a number is from zero on the number line, regardless of direction. When simplifying radical expressions, absolute values often come into play because the square root of a squared variable can be positive or negative, but we generally consider only the non-negative value in this context.
  • For example, the expression \(|x|\) is always either positive or zero, representing the distance between \(x\) and zero without considering direction.
  • In algebraic expressions, this concept allows us to express answers that remain in balance or symmetry, ensuring we capture the correct non-negative root.
In the context of simplifying \(\sqrt{z^2}\), the result is \(|z|\) because multiplying \(z\) with itself cancels the sign. Therefore, using absolute values gives us control to define this clearly.
Also, remember, nothing happens to the variables already under a radical sign unless they are perfect squares.
Exploring Square Roots
A square root takes a number and finds a value that, when multiplied by itself, gives the original number. In symbol form, \(\sqrt{a}\) is a value which, when squared, equals \(a\). Calculating square roots is tricky if you can't find a number that multiplies by itself evenly, especially with non-perfect squares. However, when simplifying expressions, identifying perfect squares simplifies the task significantly.
  • For example, \(49\) is a perfect square because \(7 \times 7 = 49\), so \(\sqrt{49} = 7\).
  • The key to working with roots is breaking down expressions into their prime factors or identifying perfect squares, simplifying where possible.
In our expression, \(\sqrt{49 z^5}\), we utilized the perfect square factorization where \(49\) itself told us \(7\).Understanding this makes simplifying easier and more logical.
Handling Variables in Radicals
Variables can sometimes make radicals more challenging. Still, the principles remain consistent: seek to simplify where you can using the properties of exponents and understanding the concept of roots. A variable in a radical represents uncertainty if the variable doesn鈥檛 form a perfect square or perfect power. Thus, breaking down the variable as much as possible aids in simplification.
  • Take \(z^5\) in our expression: breaking it as \(z^4 \times z\) allowed us to pair up and extract \(z^2\) as \(|z^2|\), offering a reduced expression.
  • Variables that can't completely escape the radical must remain under the sign, like \(z\) itself in \(\sqrt{z}\).
This maintains the mathematical integrity of the expression while making it less complex for calculations. Approaching radicals in this way helps build a robust understanding, critical for advanced algebra or calculus where radicals meet functions.

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Most popular questions from this chapter

Find the distance between the two points. Round your answer to two decimal places, if necessary. $$(-5,4),(3,-2)$$

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Writing Explain how the Distributive Property can be used to add or subtract like radicals. Give an example.
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