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Chapter 7: Problem 38

Rational Exponents Write an equivalent expression using exponential notation. $$\sqrt{22}$$

Short Answer

Expert verified
\(22^{1/2}\)

Step by step solution

01

- Identify the Radical Form

Understand that the given expression is a square root: \( \sqrt{22} \).
02

- Understand the Relationship Between Radicals and Exponents

Recall that the square root can be written as an exponent: \(\sqrt{a} = a^{1/2}\).
03

- Rewrite the Expression

Use the relationship to rewrite \(\sqrt{22}\) as: \(22^{1/2}\).

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Key Concepts

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

Exponential Notation
Exponential notation is a way of expressing numbers using a base and an exponent. It looks like this: baseexponent. For example, in the expression 3鈦�, 3 is the base, and 4 is the exponent. This means that you are multiplying the base by itself 4 times: 3脳3脳3脳3 = 81. Exponents are helpful in simplifying expressions. They show repeated multiplication in a compact form. In the context of rational exponents, you might see fractional exponents like 221/2. This is still exponential notation and means taking the base to the power of the fraction. In our exercise, the base is 22 and the exponent is 1/2.
Radicals
Radicals are a way of representing roots in mathematics, with the most common being the square root (鈭�), but you can also have cube roots (鈭�) and beyond. A square root is one that looks like this: 鈭�(a), which asks the question: 'What number, when multiplied by itself, gives you a?' For example, 鈭�(9) = 3 because 3 脳 3 = 9. It is put in simpler terms by using its equivalent fractional exponent form. For square roots, 鈭�(a) is the same as writing a1/2. This can make it easier to perform operations on the number.
Square Root
A square root is one of the most frequently encountered radicals. It is often written with a radical symbol (鈭�). For instance, 鈭�22 means you are looking for a number which, when squared (multiplied by itself), equals 22. Mathematically, it鈥檚 expressing the relationship between radicals and exponents. It's essential to understand that 鈭歛 is equivalent to a raised to the power of 1/2. This helps convert difficult square root problems into simpler forms using fractional exponents. For example, 鈭�22 can be rewritten as 221/2. This transformation helps you apply rules regarding exponents to simplify or solve equations.

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Most popular questions from this chapter

Find a simplified form for \(f(x) .\) Assume \(x \geq 0\) $$ f(x)=\sqrt{x^{3}-x^{2}}+\sqrt{9 x^{3}-9 x^{2}}-\sqrt{4 x^{3}-4 x^{2}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[3]{x^{2} y}(\sqrt{x y}-\sqrt[5]{x y^{3}}) $$

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