Problem 37 - Algebra 2

Question

Simplify each number. $$10,000^{0.75}$$

Step-by-Step Solution

Answer

1,000

Step 1: Rewrite The Fractional Exponent

Rewrite \(10,000^{0.75}\) as \(\sqrt[4]{10,000}^3\), which is the cube of the fourth root of 10,000.

Step 2: Calculate The Fourth Root

Calculate the fourth root of 10,000 which is 10. So, we have \(10^3\).

Step 3: Evaluate The Power

Lastly, evaluate the expression \(10^3\), which is 1,000.

Key Concepts

Fractional Exponents

Fractional exponents might sound complicated, but they are simply a certain way of expressing roots of numbers and are related to both roots and powers. When you see something like \(a^{n/m}\), it means that you are working with a fraction as the exponent. The denominator, \(m\), tells you the kind of root to take (such as a square root, cube root, etc.), while the numerator, \(n\), indicates the power to raise the result to.

In the example \(10,000^{0.75}\), 0.75 can be rewritten as a fraction: \(\frac{3}{4}\). Here, the number 3 is the numerator, which tells us to take the cubed power after dealing with the root, and 4 is the denominator, suggesting we take the fourth root. This method helps in simplifying expressions more efficiently as it combines roots and powers in a compact form.

Roots and Powers

Understanding roots and powers is crucial for working with fractional exponents, especially when finding the root and then raising to a power. In our step-by-step problem \(10,000^{0.75}\), we learned to first handle the roots. This involves finding the fourth root of 10,000.

• The fourth root of a number \(x\) is a number that, when raised to the fourth power, gives back \(x\). For 10,000, the fourth root is 10 because \(10^4 = 10,000\).

With the root determined, the next phase involves applying the power. After finding that the fourth root of 10,000 is 10, we then raise this result to the third power. Raising 10 to the third power, \(10^3\), means multiplying 10 by itself three times, resulting in 1,000.

Roots and powers are fundamental tools, providing flexibility in manipulating and simplifying complex mathematical expressions using fractional exponents.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form, making them easier to work with or understand. With fractional exponents, this often involves breaking down a problem into manageable steps.

In the exercise \(10,000^{0.75}\), while it initially appears complicated, it becomes manageable by understanding and applying rules for roots and powers. Following the steps:

• Rewrite the expression using fractional exponents to clearly show the root and power needed: \(\sqrt[4]{10,000}^3\).
• Calculate the fourth root first: \(\sqrt[4]{10,000} = 10\).
• Raise this result to the power of 3: \(10^3 = 1,000\).

By approaching the problem piece by piece, using fractional exponents to guide the process, we simplify what seems challenging into something more approachable. This skill of breaking down and simplifying is valuable in mathematics, streamlining problem-solving and making seemingly difficult tasks easier to handle.