Problem 87 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result of \(125^{\frac{2}{3}}\) is 25.

Step 1: Understanding the Fractional Exponent

Rewrite \(125^{\frac{2}{3}}\) as \((125^2)^{\frac{1}{3}}\). This is because \(a^{\frac{n}{m}}\) can be rewritten as \((a^n)^{\frac{1}{m}}\), which in this case makes the calculations easier.

Step 2: Square 125

Calculate 125 squared. This gives a result of 15625.

Step 3: Apply the Cubic Root

Finally, find the cubic (which is the third) root of 15625. This results in 25.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is written as \(a^n\), where \(a\) is the base and \(n\) is the exponent. The process means multiplying the base, \(a\), by itself for \(n\) number of times. For example, \(2^3\) signifies multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\). This concept is central to the solution of the problem, where understanding and manipulating bases with fractional exponents become key.
In exercises like these, you might encounter expressions with fractional exponents, such as \(125^{\frac{2}{3}}\), which can be a bit tricky at first, but follows logical steps related to regular exponentiation.

Radicals and Roots

Radicals and roots are closely related mathematical concepts. A root is essentially the inverse operation of exponentiation. For instance, the square root of a number \(a\) is a number \(b\) such that \(b^2 = a\). Radicals are symbols used to denote roots, with the radical sign \(\sqrt{}\) being the most common.

• Square roots take a number and find another number which, when squared, gives the original number.
• Cubic roots do the same, but with cubes. For example, the cubic root of 27 is 3, because \(3^3 = 27\).

In the exercise, after finding \(125^2 = 15625\), the cubic root operation is applied because of the fractional exponent \(\frac{2}{3}\), which means: first square, then take the cubic root.

Mathematical Simplification

Mathematical simplification is the process of rewriting an expression to its simplest form. This often involves operations such as reducing fractions, combining like terms, or transforming expressions. Simplification makes a complicated problem more manageable by breaking it down into easier parts.
When dealing with fractional exponents like \(125^{\frac{2}{3}}\), simplification can involve rewriting the expression in a form that follows the order of operations more straightforwardly.

• One helpful identity here is \(a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}\).
• The process allows you to separate the powers and roots, thus reducing complexity.

The solution clearly illustrates this by first squaring the base (resulting in 15625) and then finding its cubic root, effectively simplifying \(125^{\frac{2}{3}}\) to a final result of 25.