Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 5(81)^{\frac{1}{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to 15.

Step 1: Understand the Expression

The given expression is \(5(81)^{\frac{1}{4}}\). We need to express this in the simplest rational form. The expression contains a power of 81, which is raised to the \(\frac{1}{4}\) power.

Step 2: Simplify the Power

We know that \((81)^{\frac{1}{4}}\) is the same as finding the fourth root of 81. To simplify \((81)^{\frac{1}{4}}\), we determine the fourth root of 81.

Step 3: Evaluate the Fourth Root

Identify the fourth root of 81. Recall that 81 can be written as \(3^4\), because \(3 \times 3 \times 3 \times 3 = 81\). Therefore, \((81)^{\frac{1}{4}} = 3\).

Step 4: Multiply by the Coefficient

Now that we know \((81)^{\frac{1}{4}} = 3\), we multiply by the coefficient 5. Therefore, \(5 \times 3 = 15\).

Step 5: Write the Final Answer

The simplest form of the expression \(5(81)^{\frac{1}{4}}\) is 15.

Key Concepts

Understanding Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, with a non-zero denominator. This means both whole numbers and fractions fall into this category. To better understand this concept, consider these points:

• The number 4 is rational because it can be written as \( \frac{4}{1} \).
• A fraction like \( \frac{3}{5} \) is also rational because it is a ratio of two integers.
• Rational numbers include integers, fractions, and terminating or repeating decimals.

With this in mind, when asked to simplify expressions to a rational number, we aim to express them in a clear fractional form. The exercise where we're turning \(5(81)^{\frac{1}{4}}\) into a rational number ensures the final result is an integer, which in this case ends up being 15.

Unveiling the Fourth Root

The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. This operation is part of the more general concept of roots, just like the well-known square root, but involving four identical factors.

• In our exercise, we explored the fourth root of 81.
• To find the fourth root, consider: 81 can be rearranged as \(3^4\), since \(3 \times 3 \times 3 \times 3 = 81\).
• Therefore, \((81)^{\frac{1}{4}}\) simplifies to 3.

By understanding root operations, especially with powers of numbers, expressions become simpler to handle, as seen in transforming \((81)^{\frac{1}{4}}\) to 3.

The Role of Coefficients in Expressions

Coefficients are the numerical factors in algebraic expressions that multiply a variable or a power. They are pivotal in carrying out multiplication after simplifying terms such as roots.

• In our example, the coefficient before the power expression was 5.
• After establishing that \((81)^{\frac{1}{4}} = 3\), we next multiply by the coefficient: \(5 \times 3\).
• This results in the simplified rational form of 15.

Coefficients help scale the roots or variables in expressions to reach a final numerical outcome. Their application in simplifying algebraic expressions ensures that all parts of the equation work together to produce a concise result.