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Chapter 10: Problem 16

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ 25^{1 / 2} $$

Short Answer

Expert verified
The simplified form of \(25^{1/2}\) is 5.

Step by step solution

01

Understand the Given Expression

The given expression is in exponential form: \(25^{1/2}\). The exponent \(1/2\) represents a power that can be rewritten as a radical.
02

Rewrite the Expression in Radical Notation

To convert the expression to radical notation, recall that \(a^{1/n} = \sqrt[n]{a}\). Therefore, \(25^{1/2}\) can be rewritten as \(\sqrt{25}\).
03

Simplify the Radical Expression

Next, find the square root of 25. Since \(5 * 5 = 25\), \(\sqrt{25} = 5\). Thus, the simplified form of the expression is 5.

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Key Concepts

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

exponential form
In mathematics, the exponential form is a way of expressing numbers using exponents. An exponent tells us how many times a number, known as the base, is multiplied by itself.
For instance, in the expression \(25^{1/2}\), the number 25 is the base, and \(1/2\) is the exponent.
When the exponent is a fraction, it can be interpreted in terms of roots. The fraction \(1/2\) signifies the square root of the base. Therefore, \(25^{1/2}\) is the square root of 25.
Remembering these basics can help simplify many mathematical tasks and make understanding more complex equations easier.
simplifying radicals
Simplifying radicals involves reducing a radical expression to its simplest form. To simplify a radical:
  • First, identify the radical notation. For example, \(\sqrt{25}\).
  • Second, find the value that, when multiplied by itself, equals the number under the radical sign. For \(\sqrt{25}\), this number is 5 since \(5 \times 5 = 25\).
  • Thus, \(\sqrt{25}= 5\).
The goal of simplifying radicals is to make the expression as simple as possible to understand or use in further calculations.
square root
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, both 5 and -5 are square roots of 25 because:
  • \(5 \times 5 = 25\)
  • \((-5) \times (-5) = 25\)
However, in most contexts, when we talk about the square root, we mean the principal (or non-negative) square root.
Hence, the square root of 25 is referred to as 5.
To find the square root of a number, look for a value that, when squared, equals the original number under the square root sign. This process is straightforward for perfect squares (like 25).
For non-perfect squares, approximations or calculators are often used.

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Most popular questions from this chapter

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[5]{a^{6} b^{8} c^{9}} $$

Let \(f(x)=x^{2} .\) Find each of the following. $$f(7+\sqrt{3})$$

A function \(g\) is given by $$ g(z)=\frac{z^{4}-z^{2}}{z-1} $$ Find \(g(5 i-1)\)

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{\frac{1}{x+1}-\frac{2}{x}}{\frac{3}{x}+\frac{1}{x+1}}[7.5] $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{9} \sqrt[3]{3} $$
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