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Chapter 10: Problem 48

Simplify completely. Assume all variables represent positive real numbers. $$\sqrt{a^{2}}$$

Short Answer

Expert verified
The simplified expression for \(\sqrt{a^2}\) is \(a\).

Step by step solution

01

Recall properties of square roots

We have the expression \(\sqrt{a^2}\). Recall that the square root of a number squared is the number itself, i.e., \(\sqrt{x^2} = x\). So, when we apply the square root function, we are essentially undoing the power of 2.
02

Apply the square root property

Apply the square root property to the expression \(\sqrt{a^{2}}\): Since \(\sqrt{x^2} = x\), we have \(\sqrt{a^2} = a\).
03

Final answer

The simplified expression for \(\sqrt{a^2}\) is \(a\), given that \(a\) is a positive real number.

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Key Concepts

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

Simplifying Radicals
Simplifying radicals involves breaking down the root expression into its simplest form. When simplifying cube roots, like \( \sqrt[3]{54 y^{10} z^{24}} \), follow these steps:
  • Break down each term inside the radical into factors that are perfect cubes. Perfect cubes are numbers like 1, 8, 27, 64, etc., because they can be expressed as some integer raised to the power of 3.
  • For \( 54 \), its prime factors are \( 2 \times 3^3 \), where \( 3^3 = 27 \) is a perfect cube.
  • The variables \( y^{10} \) and \( z^{24} \) should also be divided into cubes. For example, \( y^{10} \) can be broken down as \( y^9 \times y \), with \( y^9 \) being \( (y^3)^3 \), a perfect cube.
  • Each term that is a perfect cube can be taken out of the radical by reducing the power by a factor of 3.
This allows you to rewrite the expression in a simpler form without a radical if possible, or with a reduced radical expression.
Properties of Exponents
The properties of exponents are essential when simplifying expressions that involve radicals and powers. They help in both breaking down and combining terms efficiently. Understand these key properties:
  • Product of Powers: \( a^m \times a^n = a^{m+n} \). This rule helps in combining powers when multiplying like bases.
  • Power of a Power: \( (a^m)^n = a^{m \times n} \). This rule is useful for simplifying expressions inside radicals, especially when dealing with cube roots.
  • Power of a Product: \( (ab)^n = a^n b^n \). This allows you to distribute a power to multiple factors inside a radical, making simplification easier.
These properties help decompose expressions like \( y^{10} \) into \( y^9 \times y \) to identify perfect cubes easily.
Positive Real Numbers
When working with real numbers, assumptions about positivity are crucial, especially for roots. Here鈥檚 why positive real numbers make simplification straightforward:
  • Non-negative Solutions: The cube root of a positive number is always positive, ensuring simpler computations.
  • Avoiding Complex Numbers: Assuming variables are positive avoids introducing imaginary numbers into the solution, which can complicate the process unnecessarily.
  • Unambiguous Roots: When numbers are positive, the principal root is straightforward to determine without needing to evaluate multiple possibilities.
This assumption not only simplifies the mathematics but also ensures the solution remains focused only on real numbers, as in the original exercise.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[4]{\frac{5}{9}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{b-25}{\sqrt{b}-5}$$

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Simplify completely. $$\frac{-10-\sqrt{50}}{5}$$
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