Textbook Question
In Exercises 75–92, rationalize each denominator. Simplify, if possible.15----------√6 + 1
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0. Review of Algebra
Radical Expressions
2:06 minutesProblem 77
Textbook Question
Evaluate each expression. (-64/27)1/3
Verified step by step guidance1
Recognize that the expression \(\left( -\frac{64}{27} \right)^{\frac{1}{3}}\) represents the cube root of the fraction \(-\frac{64}{27}\).
Recall the property of exponents that allows you to take the cube root of a fraction by taking the cube root of the numerator and the denominator separately: \(\left( \frac{a}{b} \right)^{\frac{1}{3}} = \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}\).
Apply this property to rewrite the expression as \(\frac{(-64)^{\frac{1}{3}}}{27^{\frac{1}{3}}}\).
Find the cube root of the numerator \(-64\). Since \(-64\) is a perfect cube, identify the number which when cubed gives \(-64\).
Find the cube root of the denominator \$27\(. Since \)27\( is a perfect cube, identify the number which when cubed gives \)27$.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Exponents
Rational exponents represent roots and powers combined. An expression like a^(m/n) means the nth root of a raised to the mth power. For example, a^(1/3) is the cube root of a, and a^(2/3) is the cube root of a squared.
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Cube Roots of Negative Numbers
The cube root of a negative number is also negative because cubing a negative number results in a negative number. For instance, the cube root of -64 is -4 since (-4)^3 = -64. This differs from even roots, which are not defined for negative numbers in real numbers.
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Simplifying Fractional Expressions
When dealing with fractional bases like (-64/27), apply the root to both numerator and denominator separately. For example, the cube root of (-64/27) equals the cube root of -64 divided by the cube root of 27, simplifying the expression step-by-step.
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