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Chapter 8: Problem 21

Write as a radical and then simplify. \(251 / 2\)

Short Answer

Expert verified
The simplified radical is 5.

Step by step solution

01

Interpret the Expression

To convert the expression \(25^{1/2}\) to a radical, we interpret the fractional exponent. The expression \(a^{1/b}\) corresponds to the b-th root of \(a\). Here, \(25^{1/2}\) means the square root of 25.
02

Write as a Radical Expression

Convert the expression from exponential to radical form. \(25^{1/2}\) as a radical is written as \(\sqrt{25}\). This signifies the square root of the number 25.
03

Simplify the Radical

Find the square root of 25. Since 25 is a perfect square, its square root is an integer value. Calculate \(\sqrt{25} = 5\).

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Key Concepts

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

Fractional Exponents
Fractional exponents are a way to represent roots using exponents. They are a powerful tool that allows us to write roots in a concise form. When you see an expression like \(a^{m/n}\), it indicates the nth root of \(a\) raised to the power of \(m\). Here is how we interpret this:
  • The denominator (n) of the fractional exponent indicates the root to be taken.
  • The numerator (m) indicates the power to which the root is raised.
For example, \(8^{2/3}\) means find the cube root of 8 and then square it. For \(25^{1/2}\), the exponent \(1/2\) means you take the square root of 25.
This method of expression is very useful in algebra and calculus, as it can simplify the manipulation of equations involving roots.
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. The square root is one of the most common radical expressions, and it is symbolized by the radical sign \(\sqrt{}\).
In our example, when we have \(\sqrt{25}\), it means we are looking for a number that multiplies by itself to make 25.
Key characteristics of square roots include:
  • If a number is a perfect square, like 25, the square root is a whole number.
  • If it is not a perfect square, the square root is often an irrational number.
Learning how to find square roots helps in simplification of algebraic expressions and in solving quadratic equations, adding a basic yet vital skill in math problem-solving.
Simplifying Radicals
Simplifying radicals involves reducing the expression to its simplest form. In our example, simplifying \(\sqrt{25}\) just means determining the square root of 25, which is 5.
Here are steps to simplify a radical:
  • Identify if the number inside the radical is a perfect square. If so, its square root will be an integer.
  • If it's not a perfect square, try to factor it into smaller perfect squares. For example, \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\).
Simplifying radicals is particularly important in mathematics because it makes expressions easier to work with. Avoiding long decimals and making numbers easier to manage can greatly aid in making complex algebraic calculations more approachable.

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Most popular questions from this chapter

The time, \(t\), in seconds that an object is in free fall is given by the formula \(t=s \sqrt{4}\) where s represents the distance it has fallen in feet. Calculate the time it takes an object to fall, given the following distances. Give the exact value and the approximate value rounded off to the nearest tenth of a second. $$ 192 \text { feet } $$

Perform the operations and simplify. Leave answers in exponential form. \((81 / 2) 2 / 3\)

Simplify. (Assume all radicands containing variable expressions are positive.) \(5 n 27 m n 2-----\sqrt{+} 212 m n 4------\sqrt{-} n 3 m n 2-----\sqrt{1}\)

Simplify. \(354--\sqrt{3}+5250---\sqrt{3}-416--\sqrt{3}\)

What is the square root of 1 , and what is the cube root of 1 ? Explain why.
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