Problem 12 - Algebra 1: Concepts and Skills

Question

Rewrite the expression using rational exponent notation. $$ (\sqrt[3]{5})^{2} $$

Step-by-Step Solution

Answer

The expression \((\sqrt[3]{5})^{2}\) using rational exponent notation is \(5^{\frac{2}{3}}\).

Step 1: Understand the Expression

Given is the expression \((\sqrt[3]{5})^{2}\). This sentence can be read as 'cube root of 5 all squared.' In terms of exponents, taking the cube root is the same as raising to the power of \(\frac{1}{3}\), and squaring is the same as raising to the power of 2.

Step 2: Apply the Exponentiation Rule

The rule indicates that when taking a power of a power, you multiply the exponents. Therefore, raise 5 to the power of \(2*\frac{1}{3}\), which equals \(\frac{2}{3}\). So \((\sqrt[3]{5})^{2} = 5^{\frac{2}{3}}\).

Step 3: Final Expression

So, the expression \((\sqrt[3]{5})^{2}\) in rational exponent notation is \(5^{\frac{2}{3}}\)

Key Concepts

Cube Root

Understanding the cube root is essential when dealing with rational exponent notation. The cube root of a number, which is written as \(\root{3}{x}\), is that value which, when multiplied by itself three times, equals \(x\). For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). So when we see the expression \(\root{3}{5}\), we're looking for a number that can be multiplied by itself twice more to produce 5. While the cube root of 5 doesn't result in an integer, the concept still applies when we're working with rational exponents. In rational exponent notation, a cube root is expressed as a fractional exponent: \(5^{\frac{1}{3}}\). This new notation simplifies the expression without altering its value.

Exponentiation Rule

The rules of exponentiation are fundamental in simplifying expressions with exponents. One important rule is the power of a power rule, which states that when you raise a number to an exponent and then to another exponent, you can multiply the exponents together to find the final exponent. For instance, \((x^{a})^{b} = x^{a*b}\). This rule allows us to simplify expressions like \((\root{3}{5})^{2}\) to \(5^{\frac{2}{3}}\). So when we have a cube root (or any root) being raised to a power, we can apply this rule of exponentiation to make the expression more concise and easier to work with in further calculations.

Rational Exponents

Rational exponents are a way to express roots and powers using fractions. When you convert a radical expression like \(\root{n}{x}\) to an exponent form, it becomes \(x^{\frac{1}{n}}\). If that expression is then raised to the m-th power, as in our example, it becomes \(x^{\frac{m}{n}}\). Rational exponents make it easier to perform operations such as multiplication and division with roots and powers because they follow the same rules as integer exponents. Recognizing how to work with rational exponents allows students to handle more complex equations and simplifies the process of manipulating algebraic expressions.

Expressions and Equations

Expressions and equations are the building blocks of algebra. An expression is a mathematical phrase that can contain numbers, variables, and operations, but it doesn't have an equals sign like an equation does. Expressions can be simplified using rules of arithmetic and algebra, including those governing exponentiation. An equation, on the other hand, states that two expressions are equal, setting up a problem that we can solve. When we're working with expressions involving rational exponents, we must be careful to follow all the rules that apply to exponents when simplifying or solving equations. Mastering the manipulation of expressions and understanding the structure of equations is crucial in algebra and higher levels of math.