Problem 87 - Algebra 2

Question

Find each indicated root if it is a real number. $$ \sqrt[4]{810,000} $$

Step-by-Step Solution

Answer

The fourth root of 810,000 is 30.

Step 1: Understand the root notation

The expression \(\sqrt[4]{810,000}\) refers to the fourth root of 810,000. This means we must find a number that, when raised to the power of 4, will give us 810,000.

Step 2: Calculate the fourth root

To find the fourth root of 810,000, we simply raise 810,000 to the power of \(1/4\). So the calculation will be \(810,000^{1/4}\). This can be done by using a scientific calculator.

Step 3: Check the result

After calculating \(810,000^{1/4}\), we check if the result, when raised to the power of 4, gives us back 810,000. If it does, then we know our result is correct.

Key Concepts

Root Notation

When you see a symbol like \( \sqrt[4]{810,000} \), it’s referring to what we call "root notation". This particular example asks for the fourth root of 810,000.
Imagine asking what number, multiplied by itself four times (raised to the power of 4), equals 810,000. That's the essence of finding the fourth root.

Root notation extends beyond just fourth roots, as it can represent square roots (with the number 2 usually being implicit), cube roots, or any higher-order roots.

• The number at the "index" of the root—here, 4 in \( \sqrt[4]{ } \)— tells us how many times the root must be multiplied by itself to recover the original number.
• It indicates the inverse mathematical operation of raising a number to a power or "exponent".

Understanding root notation helps us to solve equations involving roots in a structured and reliable way.

Exponents

Exponents are the flip side of roots in mathematics. They are a shorthand way to express repeated multiplication. For example, \( a^2 \) means \( a \times a \) and \( a^3 \) means \( a \times a \times a \).

In the context of finding roots, exponents become very useful.

• For the fourth root, you use an exponent of \( 1/4 \). This is because raising to the \( n \)-th root is equivalent to raising a number to the power of \( 1/n \).
• Thus, \( 810,000^{1/4} \) is used to represent \( \sqrt[4]{810,000} \).

This notation allows us to calculate roots using the same principles as calculating powers. By converting the root to an exponent form, we maintain a consistent method for handling both roots and powers in equations.
With practice, this notation becomes a versatile tool for simplifying and solving mathematical problems.

Scientific Calculator

Using a scientific calculator makes solving problems involving roots and exponents much more straightforward. These calculators are designed to handle complex calculations with minimal hassle.

When dealing with a problem like \( 810,000^{1/4} \), a scientific calculator can compute this easily. Here’s how you could approach it:

• First, enter "810000" into your calculator.
• Look for a button that either says "\( x^y \)" or has a similar exponent function.
• Enter \( 1/4 \) as the exponent to find the fourth root.
• The calculator will then provide the root, which you can verify by raising the result back to the fourth power to check your work.

These steps are typical for solving higher-order roots, and with a bit of practice, using a scientific calculator for such tasks becomes second nature.
It's essential for more advanced mathematics where manual calculations would be time-consuming and error-prone.