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Chapter 8: Problem 40

Simplify using absolute values as necessary. (a) \(\sqrt{100 y^{2}}\) (b) \(-\sqrt{100 m^{32}}\)

Short Answer

Expert verified
(a) 10 |y| , (b) -10 m^{16}

Step by step solution

01

Simplify the square root expression

First, take the square root of the constant and the variable separately for each expression. For (a) \(\sqrt{100 y^{2}} = \sqrt{100} \cdot \sqrt{y^{2}} \) and for (b) \(-\sqrt{100 m^{32}} = -\sqrt{100} \cdot \sqrt{m^{32}} \).
02

Calculate the square roots

Next, calculate the square roots of the constants and variables. \(\sqrt{100} = 10 \) so the expressions become: \( (a) \sqrt{100 y^{2}} = 10 \cdot \sqrt{y^{2}} \) and \((b) -\sqrt{100 m^{32}} = -10 \cdot \sqrt{m^{32}} \).
03

Simplify the square roots of the variables

Now simplify the square roots of the variables. Since \(\sqrt{y^{2}} = |y| \) and \(\sqrt{m^{32}} = m^{16} \), the expressions become: \((a) 10 |y| \) and \((b) -10 m^{16} \).

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Key Concepts

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

Understanding Square Roots
Square roots are mathematical operations that are the inverse of squaring a number. When you take the square root of a number, you're looking for a value that, when multiplied by itself, gives you the original number. For example, since \(10 \times 10 = 100\), it follows that \( \sqrt{100} = 10 \).

Square roots apply to both constants and variables. In the case of variables, the result often involves an absolute value. For instance, \( \sqrt{y^2} = |y| \). The absolute value ensures that the result is always non-negative, which is necessary because square roots by definition are non-negative. This non-negativity is crucial for maintaining mathematical consistency.
Role of Absolute Values
Absolute values are used to express the non-negativity of numbers and variables. The symbol \(|x|\) denotes the absolute value of \(x\), which represents the distance of \(x\) from zero on the number line, regardless of direction. For example, \(|-3| = 3\) and \(|3| = 3\).

In the context of square roots, the absolute value becomes important when you're dealing with variables. The result of a square root must be non-negative, and the absolute value notation ensures this. As explained earlier, \( \sqrt{y^2} = |y| \), not simply \(y\), because \(y\) could be negative, and the principal square root cannot be.
Understanding Exponents
Exponents are a shorthand way of indicating repeated multiplication of a number by itself. For example, \(3^4\) means \(3 \times 3 \times 3 \times 3 = 81\). When dealing with square roots and exponents, it's helpful to know that taking the square root of an exponent is equivalent to halving the exponent. For example, \( \sqrt{m^{32}} = m^{16} \).

This rule simplifies expressions. In the example given, \( \sqrt{m^{32}}\) is simplified directly to \( m^{16} \). This simplification rule is useful in various mathematical contexts, from algebra to calculus.

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Most popular questions from this chapter

Simplify. (a) \(\quad 3 \sqrt{5}+6 \sqrt{5}\)(b)\(9 \sqrt[3]{a}+3 \sqrt[3]{a}\) (c) \(5 \sqrt[4]{2 z}+\sqrt[4]{2 z}\)

Simplify. (a) \(\sqrt{72 a^{5}}-\sqrt{50 a^{5}}\)(b)\(9 \sqrt[4]{80 p^{4}}-6 \sqrt[4]{405 p^{4}}\)

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Simplify. (a) \(8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c}\)(b)\(2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}\)
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