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Chapter 8: Problem 34

Simplify using absolute values as necessary. (a) \(\sqrt{a^{12}}\) (b) \(\sqrt{b^{26}}\)

Short Answer

Expert verified
\(\sqrt{a^{12}} = \|a^6\| \) and \(\sqrt{b^{26}} = \|b^{13}\|\).

Step by step solution

01

Understand the square root property

The square root of an expression squared is equal to the absolute value of the expression. Mathematically, \(\forall x \to \sqrt{x^2} = \|x\|\). This property will help in simplifying the expressions.
02

Simplify \(a^{12}\)

Recognize that \(a^{12} = (a^6)^2\). Therefore, \(\sqrt{a^{12}} \) can be simplified using the square root property: \sqrt{a^{12}} = \sqrt{(a^6)^2} = \|a^6\|.
03

Simplify \(b^{26}\)

Recognize that \(b^{26} = (b^{13})^2\). Therefore, \(\sqrt{b^{26}} \) can be simplified using the square root property: \sqrt{b^{26}} = \sqrt{(b^{13})^2} = \|b^{13}\|.

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Key Concepts

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

Absolute Value
The absolute value of a number or expression is its distance from zero on the number line, regardless of direction. We denote it using vertical bars, like this: \(\rightarrow |x|\).

Key Points:
  • Absolute value is always non-negative.
  • It extracts the magnitude of a number without considering its sign.
  • For example, \(|-5| = 5\), and \(|5|=5\).
When we are working with square roots, we often need the absolute value because square roots only yield non-negative results even when the original number before the square was negative.

Example Problem Analysis:
In the given exercise, to simplify \(\sqrt{a^{12}}\) and \(\sqrt{b^{26}}\), we eventually use the absolute value to ensure our final expression correctly represents this non-negative result, giving us \(|a^6|\) and \(|b^{13}|\).
Square Root Property
The square root property tells us that the square root of an expression squared is the absolute value of the expression itself. This is key for simplifying roots. Mathematically, it is expressed as: \(\forall x \to \sqrt{x^2} = |x|\).

Key Points:
  • The square root function \(\sqrt{x}\) always returns the non-negative root of x.
  • When we have an expression squared inside a square root, we can break it down to its base form wrapped in absolute values.


Example Problem Analysis:
For \(\sqrt{a^{12}}\), We can express \({a^{12}}\) as \(({a^6})^2\). Applying the square root property, we get \(\sqrt{(a^6)^2} = |a^6|\).
Similarly, for \(\sqrt{b^{26}}\), we write \(({b^{13}})^2\). Applying the same principle, we get \(\sqrt{(b^{13})^2} = |b^{13}|\).
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base multiplies itself. For example, in \({a^n}\), a is the base and n is the exponent.

Key Points:
  • Any non-zero number raised to the power of zero is 1.
  • Any number raised to the power of one is itself.
  • Multiplying like bases involves adding the exponents, e.g., \({a^m \cdot a^n = a^{m+n}}\).
  • Powers of a power require multiplying the exponents, e.g., \({(a^m)^n = a^{m\cdot n}}\).

Under this concept, we simplify powers like \({a^{12}}\) and \({b^{26}}\) by recognizing patterns that can utilize exponentiation rules.

Example Problem Analysis:
For simplifying \(\sqrt{a^{12}}\), we recognize \(a^{12} = (a^6)^2\). Applying the square root property, this leads to \(\sqrt{(a^6)^2} = |a^6|\).
For \(\sqrt{b^{26}}\), note that \({b^{26}}\) can be represented as \(({b^{13}})^2\) based on exponentiation, leading to \(\sqrt{(b^{13})^2} = |b^{13}|\).

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Most popular questions from this chapter

Simplify using absolute value signs as needed. (a) \(3+\sqrt{125}\) (b) \(\frac{15+\sqrt{75}}{5}\)

Write as a radical expression. (a) \(r^{\frac{1}{2}}\) (b) \(s^{\frac{1}{3}}\) (c) \(t^{\frac{1}{4}}\)

Write with a rational exponent. (a) \(\sqrt[7]{x}\) (b) \(\sqrt[9]{y} (c) \sqrt[5]{f}\)

Use the Quotient Property to simplify square roots. (a) \(\sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}\) \\}(b) \(\sqrt[3]{\frac{24 m^{9} n^{7}}{375 m^{4} n}}\)(c) \(\sqrt[4]{\frac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}\)

Use the Quotient Property to simplify square roots. \(\sqrt{\frac{98 r^{5}}{100}}\)
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