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Chapter 7: Problem 111

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[3]{4} \cdot \sqrt{3}\)

Short Answer

Expert verified
The simplified form is \(4^{1/3} \times 3^{1/2}\).

Step by step solution

01

- Understand the problem

We are given the expression \(\root[3]{4} \times \root{3}\). We need to simplify this expression.
02

- Rewrite Radicals in Exponential Form

To simplify, express both radicals using exponents. For \(\root[3]{4}\), we write it as \(4^{1/3}\). Similarly, for \(\root{3}\), we write it as \(3^{1/2}\). Our expression now looks like \(4^{1/3} \times 3^{1/2}\).
03

- Simplify Product of Exponents

In this case, there is no further simplification that can be made to the product \(4^{1/3} \times 3^{1/2}\). This is already in its simplest form.

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Key Concepts

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

Radical Expressions
A radical expression includes a root symbol, such as a square root or cube root. The number inside the radical is called the radicand. For example, in \(\root[3]{4}\), 4 is the radicand, and the expression means the cube root of 4. Radical expressions can be simplified by expressing them in different forms.
\[ \text{For example} \ \root[3]{4} = 4^{1/3} \]
This notation makes it easier to perform algebraic operations, such as multiplication, with these expressions.
Exponents
Exponents are a fundamental part of algebra. They indicate repeated multiplication of a base number. For instance, in \(4^{1/3}\), 4 is the base, and 1/3 is the exponent, indicating a cube root. Here are a few basic rules of exponents:
  • When multiplying two expressions with the same base, you add the exponents: \ a^m \times a^n = a^{m+n} \.
  • When raising a power to another power, you multiply the exponents: \ (a^m)^n = a^{m*n} \.
  • A negative exponent means a reciprocal: \ a^{-n} = \frac{1}{a^n} \.
Simplification Steps
Simplifying radical expressions often involves several steps:
1. **Rewrite Radicals using Exponents**: Convert each radical into an expression with fractional exponents. For example, \[ \root[3]{4} = 4^{1/3} \text{ and } \root{3} = 3^{1/2} \]
2. **Combine Like Terms and Simplify**: If you are multiplying or dividing expressions, use the laws of exponents to simplify. In the original exercise, we have the product: \[ 4^{1/3} \times 3^{1/2} \] In this particular case, no further simplification is possible, so we stop here.
  • Always make sure to express your final answer in the simplest form.
  • Ensure all variables represent positive real numbers to avoid any issues with undefined or non-real answers.

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Most popular questions from this chapter

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \frac{\sqrt[3]{t^{4}}}{\sqrt[5]{t^{4}}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{27}}{3-\sqrt{3}} $$

The following expression occurs in a standard problem in trigonometry. $$ \frac{\sqrt{3}+1}{1-\sqrt{3}} $$ Show that it simplifies to \(-2-\sqrt{3}\). Then verify, using a calculator approximation.

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{5}{3 \sqrt{r}+\sqrt{s}} $$

Find the equation of a circle satisfying the given conditions. Center: (5,-2)\(;\) radius: 4
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