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Chapter 8: Problem 125

Write with a rational exponent. (a) \(\sqrt[3]{7 c}\) (b) \(\sqrt[7]{12 d}\)(c) \(2 \sqrt[4]{6 b}\)

Short Answer

Expert verified
(7c)^{1/3}, (12d)^{1/7}, 2(6b)^{1/4}

Step by step solution

01

Convert Radical to Exponent for Part (a)

Rewrite the cube root \(\root[3]{7c}\) as an expression with a rational exponent: \(\root[3]{7c} = (7c)^{\frac{1}{3}}\).
02

Convert Radical to Exponent for Part (b)

Rewrite the seventh root \(\root[7]{12d}\) as an expression with a rational exponent: \(\root[7]{12d} = (12d)^{\frac{1}{7}}\).
03

Convert Radical to Exponent for Part (c)

Rewrite the fourth root \(\root[4]{6b}\) as an expression with a rational exponent: \(\root[4]{6b} = (6b)^{\frac{1}{4}}\). Then multiply by 2 to get \ 2(6b)^{\frac{1}{4}}\.

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Key Concepts

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

Cube Roots
The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In simpler terms, for any number \(x\), the cube root \(\root[3]{x}\) is the number that satisfies the equation \(\root[3]{x} \times \root[3]{x} \times \root[3]{x} = x\).
Cube roots can be expressed as rational exponents. For instance, the cube root of \(7c\) is written as \(\root[3]{7c} = (7c)^{\frac{1}{3}}\) in exponent form.
This is handy in simplifying various mathematical expressions and solving equations.
To convert cube roots to rational exponents, remember:
  • Cube root of \(a\) = \(\root[3]{a} = a^{\frac{1}{3}}\).
  • If the number inside the radical is a product, like \(7c\), then the rational exponent applies to each factor: \( (7c)^{\frac{1}{3}} = 7^{\frac{1}{3}} \times c^{\frac{1}{3}}\).
Seventh Roots
The seventh root of a number is the value that, when used as a factor seven times, gives the original number. For any number \(x\), the seventh root \(\root[7]{x}\) satisfies the equation \(\root[7]{x} \times \root[7]{x} \times \root[7]{x} \times \root[7]{x} \times \root[7]{x} \times \root[7]{x} \times \root[7]{x} = x\).
To express seventh roots using rational exponents, convert them similarly to cube roots. For example, the seventh root of \(12d\) can be written as \(\root[7]{12d} = (12d)^{\frac{1}{7}}\).
This conversion is especially useful in advanced mathematics for simplification purposes.
Tips for converting to rational exponents:
  • Seventh root of \(a\) = \(\root[7]{a} = a^{\frac{1}{7}}\).
  • If there are multiple factors inside the radical, each is raised to the power \(\frac{1}{7}\): \( (12d)^{\frac{1}{7}} = 12^{\frac{1}{7}} \times d^{\frac{1}{7}}\).
Fourth Roots
The fourth root of a number is a value that, when multiplied by itself three more times, gives the original number. For any number \(x\), the fourth root \(\root[4]{x}\) satisfies \(\root[4]{x} \times \root[4]{x} \times \root[4]{x} \times \root[4]{x} = x\).
Fourth roots can also be expressed as rational exponents. For instance, the fourth root of \(6b\) is expressed as \(\root[4]{6b} = (6b)^{\frac{1}{4}}\). Multiplying by a scalar, like \(2 \times \root[4]{6b}\), can be written as \(2(6b)^{\frac{1}{4}}\).
Simplifying fourth roots using rational exponents makes solving equations and other mathematical operations straightforward.
Conversion tips:
  • Fourth root of \(a\) = \(\root[4]{a} = a^{\frac{1}{4}}\).
  • For expressions with multiple factors: \( (6b)^{\frac{1}{4}} = 6^{\frac{1}{4}} \times b^{\frac{1}{4}}\).

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Most popular questions from this chapter

Why is there no real number equal to \(\sqrt{-64}\) ?

Write as a radical expression. (a) \(r^{\frac{1}{2}}\) (b) \(s^{\frac{1}{3}}\) (c) \(t^{\frac{1}{4}}\)

Simplify using absolute value signs as needed. (a) \(\sqrt{y^{11}}\) (b) \(\sqrt[3]{r^{5}}\) (c) \(\sqrt[4]{s^{10}}\)

Simplify using absolute values as necessary. (a) \(\sqrt[7]{128 r^{14}}\) (b) \(\sqrt[4]{81 s^{24}}\)

Explain why \(\sqrt[4]{-64}\) is not a real number but \(\sqrt[3]{-64}\) is.
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