Problem 77 Simplify. Assume that all variab... [FREE SOLUTION]

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Chapter 10: Problem 77

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt[3]{27 t^{12}}\)

Short Answer

Expert verified

-3t^4

Step by step solution

Identify the Cube Root

First, recognize that the expression is under a cube root: ewline ewline \[-\sqrt[3]{27 t^{12}}\]

Break Down the Expression Inside the Root

Break down the expression inside the cube root into its factors: ewline \[-\sqrt[3]{27 t^{12}} = -\sqrt[3]{27} \times \sqrt[3]{t^{12}}\]

Simplify the Cube Root of 27

Simplify the cube root of 27. Since \[27 = 3^3\], we get ewline \[-\sqrt[3]{27} = -3\]

Simplify the Cube Root of the Variable Term

Simplify the cube root of \ t^{12} \, as follows: ewline \[\sqrt[3]{t^{12}} = t^{12/3} = t^4\]

Combine the Results

Now combine the results from steps 3 and 4: ewline \[-3 \times t^4 = -3t^4\]

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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 mathematical operation. It reverses the effect of cubing a number. For example, the cube root of 8 is 2 because 2 cubed is 8. The cube root symbol is \(\sqrt[3]{...}\). It's used to find a number that, when multiplied by itself three times, equals the given number. In our problem, we have \-\(\sqrt[3]{27 t^{12}}\). We need to simplify this expression by separately taking the cube root of the numbers and variables inside.

algebraic simplification

Algebraic simplification makes expressions easier to work with by rewriting them in simpler forms. It's like cleaning up a messy room. We take complicated terms and make them neat and clear. In the given exercise, we break down the original expression \-\(\sqrt[3]{27 t^{12}}\) into smaller, manageable pieces to simplify each part. We first separate the numbers and variables inside the cube root.

exponent rules

Exponent rules help us handle mathematical operations involving exponents. They provide shortcuts for dealing with powers. For example, the power rule says that \( (a^m)^n = a^{m \times n} \). In the exercise, simplifying \(\t^{12}\) under a cube root uses exponent rules. We write \( \sqrt[3]{ t^{12} } = t^{12/3} = t^4 \). Breaking exponents into smaller parts helps us simplify complex expressions easily.

positive real numbers

Positive real numbers are all numbers greater than zero. They include both whole numbers and fractions. In algebraic simplification, we often assume variables represent positive real numbers. This assumption helps us avoid complications with negative roots and undefined values. In our example, \(\t\) is considered a positive real number, allowing us to simplify expressions involving \(\t\) and its exponents without concern for negative values.

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91影视

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt[3]{27 t^{12}}\)

Short Answer

Step by step solution

Identify the Cube Root

Break Down the Expression Inside the Root

Simplify the Cube Root of 27

Simplify the Cube Root of the Variable Term

Combine the Results

Key Concepts

cube root

algebraic simplification

exponent rules

positive real numbers

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