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Chapter 0: Problem 87

Evaluate each expression without using a calculator. $$125^{\frac{2}{3}}$$

Short Answer

Expert verified
The value of \(125^{2/3}\) is 25.

Step by step solution

01

Find the Cubic Root

Find the cubic root of 125. This is done by determining which number multiplied by itself three times equals 125. In this case, the number is 5 because \(5^3 = 125\).
02

Square the Result

Take the result found in the previous step, and square it. This means multiplying 5 by itself, which gives \(5^2=25\).

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Key Concepts

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

Cubic Root
Understanding the concept of a cubic root is crucial to solving expressions like \( 125^{\frac{2}{3}} \). The cubic root of a number is a value that, when multiplied by itself three times, gives the original number back.

For example, to find the cubic root of 125, you ask yourself, "What number, multiplied by itself three times, equals 125?" The answer is 5, because \( 5 \times 5 \times 5 = 125 \).

This step is often an essential part of dealing with expressions involving rational exponents when the denominator is 3.
Rational Exponents
Rational exponents combine both roots and powers in one expression, making them a fascinating topic. When you see an exponent like \( \frac{2}{3} \), it can be interpreted as both taking a root and raising to a power. The denominator of the fraction tells you which root to take, while the numerator tells you to which power the result should be raised.

For \( 125^{\frac{2}{3}} \), the \( \frac{1}{3} \) part instructs you to find the cubic root of 125, which is 5. Next, you raise this result to the 2nd power as indicated by the numerator. This approach connects the idea of roots and exponents seamlessly in one operation.
Powers and Roots
The relationship between powers and roots is central to simplifying expressions using rational exponents.

Powers, like squaring, mean multiplying a number by itself a certain number of times. For instance, \( 5^2 = 5 \times 5 = 25 \). Roots, on the other hand, involve finding a number which, when used repeatedly in multiplication, equals the given value.

In the expression \( 125^{\frac{2}{3}} \), both actions are necessary. First, finding the cubic root of 125 simplifies it to 5, and then squaring 5 gives us 25. By understanding this symmetry, one can better handle expressions with rational exponents and achieve the correct simplification.

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Most popular questions from this chapter

Explain how to factor the difference of two squares. Provide an example with your explanation.

Using an example, explain how to factor out the greatest common factor of a polynomial.

Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$\frac{x^{2}+6 x+5}{x^{2}-25}$$

What does it mean to factor completely?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$534.7=5.347 \times 10^{3}$$
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