Problem 955 Simplify: (a) \(^{3} \sqrt{-51... [FREE SOLUTION]

Chapter 33: Problem 955

Simplify: (a) \(^{3} \sqrt{-511}\) (b) \(^{4} \sqrt{81 / 16}\) (c) \(^{3} \sqrt{-16} \div{ }^{3} \sqrt{-2}\).

Short Answer

Expert verified

(a) \(- ^3\sqrt{511}\) (b) \(\frac{3}{2}\) (c) \(\frac{2}{^3\sqrt{2}}\)

Step by step solution

(a) Simplify the cube root of a negative number

Given expression: \( ^3\sqrt{-511} \) To simplify this expression, we first need to acknowledge that the cube root of a negative number is negative, since a negative number multiplied by itself three times will yield a negative product. In this case, we can write: \( ^3\sqrt{-511} = - ^3\sqrt{511} \) Since 511 does not have any perfect cube factor, we can leave the expression in this form: \( - ^3\sqrt{511} \)

(b) Simplify the fourth root of a fraction

Given expression: \( ^4\sqrt{81 / 16} \) To simplify this expression, we can use the property of radicals: \( \sqrt[a]{x/y} = \sqrt[a]{x} / \sqrt[a]{y} \). In this case, we can write: \( ^4\sqrt{81 / 16} = ^4\sqrt{81} / ^4\sqrt{16} \) Now, we know that 81 is 3^4 and 16 is 2^4, so we can simplify the expression as: \( ^4\sqrt{81} / ^4\sqrt{16} = 3 / 2 \) Thus, the simplified expression is \( 3 / 2 \).

(c) Simplify a cube root of a negative number divided by another cube root of a negative number

Given expression: \( ^3\sqrt{-16} \div{ } ^3\sqrt{-2} \) First, we can simplify both cube roots of negative numbers: \( ^3\sqrt{-16} = - ^3\sqrt{16} = -2 \) \( ^3\sqrt{-2} = - ^3\sqrt{2} \) Now, we can substitute the simplified expressions back into the original expression: \( -2 \div{ } (- ^3\sqrt{2}) = -2 / (- ^3\sqrt{2}) \) Since we have two negatives in the expression, we can simplify it further: \( -2 / (- ^3\sqrt{2}) = 2 / ^3\sqrt{2} \) Thus, the simplified expression is \( 2 / ^3\sqrt{2} \).

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Key Concepts

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

Cube Root

The cube root is a special type of radical expression. It involves finding a number which when multiplied by itself three times gives the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). When dealing with negative numbers, the cube root is also negative because multiplying three negative numbers results in a negative number. In mathematical terms, \( ^3\sqrt{-x} = - ^3\sqrt{x} \).

Example: The cube root of -27 is -3, since \(-3 \times -3 \times -3 = -27\).

In the exercise, to simplify \(^{3}\sqrt{-511}\), recognize that it is the same as \(- ^3\sqrt{511}\), since 511 does not contain a perfect cube factor, thus maintaining this form is appropriate.

Fourth Root

The fourth root involves finding a number that, when multiplied by itself four times, results in the original number. For instance, the fourth root of 16 is 2 because \(2 \times 2 \times 2 \times 2 = 16\).

To calculate the fourth root of a fraction, like \(^{4}\sqrt{81/16}\), break it into two separate fourth roots: \(^{4}\sqrt{81}\) and \(^{4}\sqrt{16}\).
Knowing that \(81 = 3^4\) and \(16 = 2^4\), simplifies to \(3/2\), because each of these is a perfect fourth power, simplifying the fraction inside.

Negative Numbers

Negative numbers can complicate radical expressions, especially in roots, due to their influence on the sign of a product.

Cubing a negative number remains negative, making handling cube roots of negative numbers special.
In contrast, square roots and even fourth roots of negative numbers aren't typically defined among real numbers but involve complex numbers.

When simplifying an expression like \(^{3}\sqrt{-16} \div ^{3}\sqrt{-2}\), identify each step:

\(^{3}\sqrt{-16}\) = \(-^{3}\sqrt{16}\) = -2
\(^{3}\sqrt{-2}\) = \(-^{3}\sqrt{2}\)

Putting into the expression, the negatives cancel each otherleaving \(2/^{3}\sqrt{2}\). Thus, you convert overwhelming negative expressions into simpler positive outcomes.

Fraction Simplification

Fractions often appear in radical expressions, and simplifying them requires dividing large tasks into simpler ones. When dealing with roots of fractions, remember that \(\sqrt[a]{x/y} = \sqrt[a]{x} / \sqrt[a]{y}\).

Apply this to fourth roots or cube roots by treating the numerator and denominator separately.
In \(^{4}\sqrt{81/16}\), separate to calculate \(^{4}\sqrt{81}\), which is 3, and \(^{4}\sqrt{16}\), which results in 2.

Thus, this becomes \(3/2\). Fraction simplification in radical expressions makes problems manageable, so always look to break down and simplify each part individually before combining.

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91影视

Simplify: (a) \(^{3} \sqrt{-511}\) (b) \(^{4} \sqrt{81 / 16}\) (c) \(^{3} \sqrt{-16} \div{ }^{3} \sqrt{-2}\).

Short Answer

Step by step solution

(a) Simplify the cube root of a negative number

(b) Simplify the fourth root of a fraction

(c) Simplify a cube root of a negative number divided by another cube root of a negative number

Key Concepts

Cube Root

Fourth Root

Negative Numbers

Fraction Simplification

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