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Chapter 1: Problem 49

Rewrite the expression using rational exponents. $$\sqrt[3]{(a+b)^{2}}$$

Short Answer

Expert verified
\( (a+b)^{\frac{2}{3}} \)

Step by step solution

01

Identify the Radical

The expression given is a cube root, \( \sqrt[3]{(a+b)^{2}} \). The radical used is a cube root, which implies a fractional exponent.
02

Translate the Radical

Using the property of radicals, \( \sqrt[n]{x^m} = x^{\frac{m}{n}} \), rewrite the expression: \( \sqrt[3]{(a+b)^{2}} = (a+b)^{\frac{2}{3}} \).
03

Confirm the Expression

Make sure all parts of the expression have been properly converted to rational exponents. \( (a+b)^{\frac{2}{3}} \) now effectively represents the cube root of \((a+b)^2\).

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Key Concepts

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

Understanding the Cube Root
The cube root is an important concept that you may encounter frequently in mathematics. Essentially, finding the cube root of a number is the reverse operation of cubing a number. In other words, it asks: what number, when multiplied by itself twice, gives the original number? For example, the cube root of 27 is 3, because when 3 is multiplied by itself twice (3 \(\times\) 3 \(\times\) 3), it results in 27.
  • The sign for the cube root is a radical sign with a small number 3 written just outside (\(\sqrt[3]{x}\)).
  • Cube roots can be applied to both numbers and expressions.
  • They can be expressed in the form of fractional exponents for simplicity.
Understanding cube roots helps in simplifying and solving equations where expressions are embedded with radicals. It makes calculations easier, especially when these expressions are rewritten using exponents.
Exploring Fractional Exponents
Fractional exponents provide a compact way to express roots and powers. They combine the concept of raising a number to a power and taking a root of it in a single expression.

A fractional exponent like \(x^{\frac{m}{n}}\) is interpreted as taking the \(n\)-th root of \(x\) and then raising it to the \(m\)-th power.
  • For instance, \((a+b)^{\frac{2}{3}}\) means take the cube root of \((a+b)^2\).
  • The numerator in the fractional exponent (2 in this case) is the power.
  • The denominator (3) is the root.
Fractional exponents are useful because they allow us to manipulate and simplify expressions involving roots, using the laws of exponents. They enable the transformation of seemingly complex problems into more manageable forms.
Properties of Radicals Explained
Radicals have properties that are crucial for solving problems involving square roots, cube roots, and higher-order roots. These properties simplify complex expressions and help prevent errors during calculations. Here are some key properties to keep in mind:
  • Product Property: \(\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}\). This allows you to break down complex radicands into simpler parts.
  • Quotient Property: \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\). Useful for dividing radicands within a radical.
  • Power Rule for Radicals: \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\). This rule is essential for rewriting radicals as fractional exponents.
These properties are foundational tools that transform complicated radical expressions into forms that are easier to evaluate or simplify. Using them consistently in your problem solving will aid in mastering algebra and calculus concepts more effectively.

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Most popular questions from this chapter

The average weight \(W\) (in pounds) for women with height \(h\) between 60 and 75 inches can be approximated using the formula \(W=0.1049 h^{1.7}\) Construct a table for \(W\) by letting \(h=60,61, \ldots, 75\) Round all weights to the nearest pound. $$\begin{array}{|c|c|c|c|} \hline \text { Height } & \text { Weight } & \text { Height } & \text { Weight } \\ \hline 60 & & 68 & \\ \hline 61 & & 69 & \\ \hline 62 & & 70 & \\ \hline 63 & & 71 & \\ \hline 64 & & 72 & \\ \hline 65 & & 73 & \\ \hline 66 & & 74 & \\ \hline 67 & & 75 & \\ \hline \end{array}$$

Simplify the expression. $$\frac{t}{t+3}+\frac{4 t}{t-3}-\frac{18}{t^{2}-9}$$

Evaluate the pair of expressions for \(x=1,2,3\) 4, and 5 by constructing a table of values. Discuss whether or not the two expressions might be equal. $$\frac{113 x^{3}+280 x^{2}-150 x}{22 x^{3}+77 x^{2}-100 x-350}, \frac{3 x}{2 x+7}+\frac{4 x^{2}}{1.1 x^{2}-5}$$

Simplify the expression. $$\frac{12+r-r^{2}}{r^{3}+3 r^{2}}$$

Express as a sum of terms of the form , where r is a rational number. $$\frac{(\sqrt{x}-3)^{2}}{x^{3}}$$
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