Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 10,000^{\frac{3}{4}} $$

Step-by-Step Solution

Answer

The simplest form of \(10,000^{\frac{3}{4}}\) is 1000.

Step 1: Understand the Expression

The given expression is \(10,000^{\frac{3}{4}}\). This represents a power of 10,000. The exponent \(\frac{3}{4}\) indicates that we first find the fourth root of 10,000 and then cube the result.

Step 2: Calculate the Fourth Root of 10,000

To find the fourth root of 10,000, rewrite 10,000 as \(10^4\). Thus, \(\sqrt[4]{10,000} = \sqrt[4]{10^4} = 10^{4/4} = 10^1 = 10\).

Step 3: Raise the Result to the Third Power

Now that we have \(10\) from the previous step, we need to raise it to the third power according to the exponent \(\frac{3}{4}\) which means cubing the fourth root result. Therefore, \(10^3 = 1000\).

Step 4: Express the Final Answer

The expression \(10,000^{\frac{3}{4}}\) is simplified to a rational number. The simplified form is 1000.

Key Concepts

Exponents

Exponents are a fundamental mathematical concept used to describe the operation of repeating multiplication. When you see an expression like \(a^b\), the number \(a\) is the base, and \(b\) is the exponent. This tells us how many times to multiply the base by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\). Exponents can be whole numbers, but they can also be fractions, as in the exercise provided. Fractional exponents are a bit more complex, pairing multiplication and roots together. In the exercise \(10,000^{\frac{3}{4}}\), the exponent \(\frac{3}{4}\) informs us of two operations: finding a root and then performing repeated multiplication.

Roots

Roots, quite simply, are the opposite of powers or exponents. When we talk about square roots or fourth roots, we mean the number that needs to be multiplied by itself a certain number of times to achieve the original number. The fourth root of 10,000, for instance, is a number that, when multiplied by itself four times, yields 10,000. To find the fourth root, you can express 10,000 in terms of another number raised to a power, like \(10^4\). The fourth root of \(10^4\) is \(10^{4/4} = 10\). By simplifying the exponent, you can see that the fourth root of 10,000 is 10.

Simplest Form

Simplifying a mathematical expression into its simplest form makes it easier to understand and work with. In this context, it refers to reducing a fraction or an expression with exponents to its most reduced and manageable state. For \(10,000^{\frac{3}{4}}\), the solution involves a two-step simplification: first finding the fourth root of 10,000 (which simplifies to 10), then raising this result to the third power, yielding 1,000. This reduction process helps clear up the complexity of dealing with the original fractional exponent, making the answer neat and tidy.

Rational Expressions

A rational expression is a fraction that involves polynomials in its numerator and denominator. However, in a broader view, it includes expressions that can be converted or expressed as fractions, particularly those involving roots and exponents. Rational expressions can be simplified and manipulated using similar methods as rational numbers. The exercise \(10,000^{\frac{3}{4}}\) is expressed as a rational number after simplifying the exponent and the operations it dictates, leading to the final rational number, 1,000. This transformation highlights the importance of rational expressions in converting complex mathematical forms into simpler, manageable ratios.