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Chapter 8: Problem 26

Write as a radical and then simplify. \((49) 1 / 2\)

Short Answer

Expert verified
The simplified radical form is 7.

Step by step solution

01

Understand and Rewrite as a Radical

The expression given is \((49)^{1/2}\). The exponent \(1/2\) means that we are taking the square root of 49. We can rewrite this expression using radical notation as \(\sqrt{49}\).
02

Simplify the Radical

Now that the expression is written as \(\sqrt{49}\), our task is to simplify it. We know that 49 is a perfect square because \(7 \times 7 = 49\). Therefore, \(\sqrt{49} = 7\).

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Key Concepts

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

square root
The square root is a way of finding a number that, when multiplied by itself, gives you the original number. It is often represented with the radical symbol \( \sqrt{} \). For example, \( \sqrt{49} \) indicates that we need to find a number which, when multiplied by itself, results in 49. The answer here is 7, since \( 7 \times 7 = 49 \).
  • The square root operation is fundamentally important when solving quadratic equations or simplifying expressions.
  • Finding square roots can often be simplified if the number is a perfect square.

When dealing with square roots, it's crucial to remember that every positive number has two square roots; one positive and one negative. For example, both 7 and -7 are square roots of 49, because \( 7^2 = 49 \) and \( (-7)^2 = 49 \). But usually, when we refer to the square root, we mean the positive one.
rational exponents
Rational exponents are an alternative way to express roots, using fractions as exponents. In the expression \( (49)^{1/2} \), the exponent \( \frac{1}{2} \) indicates that we're taking the square root of 49. Here, the numerator (1) tells us the power, and the denominator (2) tells us the root.
  • Expression with rational exponents can often be simplified by converting them to radical form.
  • Rational exponents follow the rules of exponents, such as \(a^{m/n} = \sqrt[n]{a^m}\).
  • They offer a compact way to denote roots and powers in mathematical expressions.

By understanding rational exponents, students are better able to manipulate and simplify complex mathematical expressions. For instance, \( (49)^{1/2} \) is straightforward as it directly translates to \( \sqrt{49} \), which simplifies to 7.
perfect squares
A perfect square is a number that is the square of an integer. In other words, when an integer is multiplied by itself, the product is a perfect square. Examples include 1, 4, 9, 16, 25, 36, 49, and so on. Perfect squares are important to identify because they simplify the process of finding square roots.
  • An easy way to identify a perfect square is to find its integer square root and multiply it by itself.
  • Recognizing perfect squares quickly can greatly aid in simplifying radical expressions.

In the example given, 49 is a perfect square because it is the result of \( 7 \times 7 \). Therefore, \( \sqrt{49} = 7 \). Recognizing that 49 is a perfect square allows us to efficiently compute its square root without the use of a calculator.

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Most popular questions from this chapter

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