Problem 33 Write as a radical and then simp... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 8: Problem 33

Write as a radical and then simplify. $(127) 1 / 3$

Short Answer

Expert verified

The expression is $\sqrt[3]{127}$ and it cannot be simplified further.

Step by step solution

Convert to Radical Form

The expression $127^{1/3}$ is an example of a number in exponential form where the exponent is a fraction. To write it as a radical, recognize that the denominator of the fraction (3) indicates the index of the radical. Therefore, $127^{1/3}$ can be written as $\sqrt[3]{127}$.

Simplify the Radical

Next, we examine $\sqrt[3]{127}$ to see if it can be simplified further. A radical expression is simplified if there are no perfect cube factors (other than 1) in the radicand. Since 127 is a prime number, the cube root of 127 cannot be simplified further.

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Key Concepts

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

Exponential Form

Exponential form is a way of expressing numbers using exponents. It is particularly useful for simplifying complex calculations. In our exercise, the expression $127^{1/3}$ is in exponential form. The base "127" is the number being multiplied, and "1/3" is the exponent, which signifies that you are taking the cube root of 127. Here are some key points to note about exponential form:

The base is the number or expression that is being raised to a power.
The exponent tells us how many times to use the base in a multiplication.
A fractional exponent, like $1/3$, indicates a root. Specifically, it tells us to take the cube root of 127 in this exercise.

Understanding the exponential form is important as it provides a concise way to represent and work with large numbers or roots.

Radical Simplification

Radical simplification involves expressing a radical expression in its simplest form. This involves factoring the radicand (the number under the radical sign) to reduce it as much as possible. However, not all radicals can be simplified further.In our exercise, we encounter $\sqrt[3]{127}$. To simplify, we look for any perfect cube factors in 127:

A perfect cube is a number that can be expressed as $n^3$ (e.g., 8 is a perfect cube since $2^3 = 8$).
Since 127 is a prime number, it has no perfect cube factors other than 1 and itself.

Thus, the radical $\sqrt[3]{127}$ is already in its simplest form, and no further simplification is possible.

Cube Root

A cube root is the number that, when multiplied by itself twice, results in the original number. It is the inverse operation of cubing a number. The cube root is denoted by the radical symbol with an index of 3, as seen in $\sqrt[3]{x}$.In the example $\sqrt[3]{127}$:

The cube root of 127 is the number which, if multiplied by itself twice, equals 127.
Cube root operations can simplify expressions where the radicand is not a perfect cube, as in this case.

While perfect cubes are easier to compute, like the cube root of 8 (which is 2), non-perfect cubes, like 127, result in an irrational number and are thus expressed in radical form.

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91影视

Write as a radical and then simplify. \((127) 1 / 3\)

Short Answer

Step by step solution

Convert to Radical Form

Simplify the Radical

Key Concepts

Exponential Form

Radical Simplification

Cube Root

One App. One Place for Learning.

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