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Chapter 7: Problem 76

Simplify each root. $$ \sqrt[4]{k^{20}} $$

Short Answer

Expert verified
The simplified form is \(\boldsymbol{k^{5}}\).

Step by step solution

01

Understand the Root and Exponent Relationship

The fourth root of a number is equivalent to raising that number to the power of \(\frac{1}{4}\). Therefore, we have \[ \sqrt[4]{k^{20}} = (k^{20})^{1/4} \]
02

Apply the Power Rule

When raising a power to another power, multiply the exponents. This gives us: \[ (k^{20})^{1/4} = k^{20 \cdot \frac{1}{4}} \]
03

Simplify the Exponent

Multiply the exponents together to simplify: \[ 20 \cdot \frac{1}{4} = 5 \] Therefore, \[ k^{20 \cdot \frac{1}{4}} = k^{5} \]

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Key Concepts

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

headline of the respective core concept
Simplifying radicals in algebra involves understanding how to work with roots and exponents. In particular, for problems like \( \sqrt[4]{k^{20}} \), we need to use exponent rules and the power rule to simplify the expression. Let鈥檚 break down the main concepts to help you master these techniques.
Fourth Roots
The fourth root of a number involves finding a value that, when multiplied by itself four times, gives the original number.
For example, the fourth root of \( 16 \) is \( 2 \) because \( 2 \times 2 \times 2 \times 2 = 16 \).
In algebraic terms, the fourth root of \( k^{20} \) can be written as \( \sqrt[4]{k^{20}} \).
Alternatively, using exponents, this is equivalent to raising \( k^{20} \) to the power of \( \frac{1}{4} \), so we have \( (k^{20})^{1/4} \).
The key here is transforming a root into an exponent to make simplification easier. Understanding this transformation helps you simplify complex expressions by applying the exponent rules.
Exponent Rules
Exponent rules are essential when working with expressions involving powers.
Some important rules include:
  • The product rule: \( a^m \times a^n = a^{m+n} \)
  • The quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \)
  • The power rule: \( (a^m)^{n} = a^{m \times n} \)
When simplifying \( (k^{20})^{1/4} \), we use the power rule.
This tells us that when raising a power to another power, we need to multiply the exponents.
So, we multiply \( 20 \times \frac{1}{4} \), which simplifies to \( 5 \).
Power Rule in Exponents
The power rule in exponents is a crucial concept whenever you raise an exponent to another exponent.
This rule states that \( (a^m)^n = a^{m \times n} \).
Let's apply this rule using our original problem: \( (k^{20})^{1/4} \).
According to the power rule, we multiply the exponents: \( 20 \times \frac{1}{4} = 5 \).
Thus, the simplified form is \( k^{5} \).
The power rule is efficient for handling nested exponents and converting roots into simpler exponential forms.
By mastering it, you can streamline complex algebraic expressions and tackle a wide range of problems with confidence.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent positive real numbers. $$ -8 y^{11 / 7}\left(y^{3 / 7}-y^{-4 / 7}\right) $$

The following expression occurs in a standard problem in trigonometry. $$ \frac{\sqrt{3}+1}{1-\sqrt{3}} $$ Show that it simplifies to \(-2-\sqrt{3}\). Then verify, using a calculator approximation.

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{5 \sqrt{k}}{2 \sqrt{k}+\sqrt{q}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{12}{\sqrt{6}+\sqrt{3}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}} $$
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