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Chapter 0: Problem 62

Simplify each expression. . \(\sqrt[5]{y^{5}}\)

Short Answer

Expert verified
y

Step by step solution

01

Understand the Radical Notation

Identify the expression inside the radical. Here, the expression is \(y^{5}\).
02

Recognize the Power and the Root

The radical \( \sqrt[5]{...} \) is the 5th root. The exponent of y is 5. So, we need to simplify \( \sqrt[5]{y^{5}} \).
03

Apply the Property of Radicals

Use the property that \( \sqrt[n]{a^n} = a \). Here, \( n = 5 \) and \( a = y\). Thus, \( \sqrt[5]{y^{5}} = y \).

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Key Concepts

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

Radical Notation
Radical notation is a way to represent roots of numbers or expressions. The radical sign is what you use to denote this. For example, the square root uses the symbol \(\sqrt{...}\) and higher roots use notation like \(\sqrt[3]{...}\) for cube roots or \(\sqrt[5]{...}\) for fifth roots.
Here is how a fifth root works: when you see \(\sqrt[5]{y^5}\), the number #5 inside the radical is the root we are taking.
The expression inside the radical (in this case, \(y^5\)) is called the radicand.
.
Exponents
Exponents are a way to express repeated multiplication. If you have \(y^5\), it means you're multiplying \(y\) by itself five times: \(y \times y \times y \times y \times y\).
Here鈥檚 a quick refresher on some rules of exponents:
  • \(a^1 = a\)
  • \(a^0 = 1\) (where \(a\) is not zero)
  • \((a^m)^n = a^{mn}\)
  • \(a^m \cdot a^n = a^{m+n}\)
Using exponents helps in simplifying expressions and solving equations efficiently.
.
Properties of Radicals
Simplifying radicals often involves using their properties. One extremely helpful property is \(\sqrt[n]{a^n} = a\). This tells us that taking the n-th root of a number raised to the n-th power returns the original number.
Here is a step-by-step on how we used this property in the exercise:
  • We had \(\sqrt[5]{y^5}\).
  • We identified 'n' as 5, so the fifth root of \(y^5\) needs to be simplified.
  • We applied the property: \(\sqrt[5]{y^5} = y\).
    • This is because the 5th root and raising to the power 5 are inverse operations.

Knowing and using this property can make radical expressions much easier to work with.
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Most popular questions from this chapter

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ \left(2 c^{3} \sqrt{d}+3 d^{3} \sqrt{c}\right)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ \left(5 a^{2} \sqrt{b}+7 b^{2} \sqrt{a}\right)^{2} $$

Perform the indicated operations and simplify. $$ \left(\frac{1}{5} c-\frac{2}{3} d^{3}\right)\left(\frac{1}{5} c+\frac{2}{3} d^{3}\right) $$

Explain the similarity in simplifying the given expressions. a. \((3 x+2)(4 x-7)\) b. \((3 \sqrt{x}+\sqrt{2})(4 \sqrt{x}-\sqrt{7})\)

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{7+3 \sqrt{z}})(\sqrt{7-3 \sqrt{z}}) $$
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