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Chapter 7: Problem 12

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

Short Answer

Expert verified
The expression simplifies to 15.

Step by step solution

01

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.
02

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.
03

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, \((81)^{\frac{1}{4}} = 3\).
04

Multiply by the Coefficient

Now that we know \((81)^{\frac{1}{4}} = 3\), we multiply by the coefficient 5. Therefore, \(5 \times 3 = 15\).
05

Write the Final Answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{3}{a^{-5}} $$

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 12^{\frac{5}{4}} $$

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{25 a} $$

Show that \(3 \times 10^{-2}=\frac{3}{100}\)

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{20 x^{0} y^{-5}}{4 x^{-1} y^{5}} $$
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