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Chapter 7: Problem 1

Simplify each expression. $$36^{\frac{1}{2}}$$

Short Answer

Expert verified
The simplified expression for \( 36^{\frac{1}{2}} \) is 6.

Step by step solution

01

Identify the operation

In the expression \( 36^{\frac{1}{2}} \), the power \( \frac{1}{2} \) means finding the square root of the base number, which is 36
02

Compute the square root

Compute the square root of 36 as such: \( \sqrt{36} \). The square root of 36 is 6.

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Key Concepts

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

Radicals
Radicals are a way to represent roots of numbers. When you see a radical sign (鈭�), it usually refers to the square root. However, other types of roots, such as cube roots or fourth roots, can also be expressed using radicals. The part under the radical is called the radicand, which is the number you're taking the root of. For instance, in the expression \( \sqrt{36} \), 36 is the radicand.

Dealing with radicals becomes relatively straightforward once you learn the basics:
  • Simplify the radicand if possible by factoring it into a product of perfect squares.
  • Use the property that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) to simplify complex radicals.
  • Remember that you can express radicals as fractional exponents. For example, \( \sqrt{36} \) can also be written as \( 36^{\frac{1}{2}} \).
Simplification
Simplifying expressions is about reducing them to their most basic form. This often involves eliminating redundant operations and converting expressions to a form that is both easier to read and easier to compute with.

In the context of exponents and radicals, simplification typically involves:
  • Recognizing and applying mathematical identities, such as \( a^{\frac{m}{n}} = n\sqrt{a^m} \), which can help switch between radicals and exponents.
  • Combining like terms, if possible.
  • Performing arithmetic operations like addition, subtraction, or multiplication on simplified forms.
For example, when you simplify \( 36^{\frac{1}{2}} \), you are finding its simplest radical form: \( \sqrt{36} = 6 \). This removes the exponent and radical, resulting in a whole number which is easier to understand and use in further calculations.
Square Root
The square root is a particular kind of radical. It involves finding a number that, when multiplied by itself, results in the original number. When you take the square root of a number \( n \), you are finding the number \( x \) such that \( x^2 = n \).

Common elements to keep in mind when working with square roots include:
  • Only non-negative numbers have real number square roots.
  • If \( n \) is a perfect square like 36, the square root is a simple integer, in this case, 6 because \( 6 \times 6 = 36 \).
  • You can simplify square roots by factoring the number into its components and finding pairs of numbers to simplify.
Square roots often appear in equations and various applications, so understanding how to compute and simplify them is crucial for solving many math problems efficiently.

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Most popular questions from this chapter

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(\sqrt{2 x-3}=4\)

Find each indicated root if it is a real number. $$ -\sqrt[4]{16} $$

Graph each function. \(y=\sqrt[3]{x-4}\)

a. Graph \(y=\sqrt{x-2}+1\) and \(y=-\sqrt{x-2}+1\) b. Find the domain and the range of each function.

What is the inverse of \(y=x^{2}-2 x+1 ?\) Is the inverse a function? Explain.
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