Problem 52 - Precalculus: Functions and Graphs

Question

Rewrite the expression using rational exponents. $$\sqrt[3]{r^{3}-s^{3}}$$

Step-by-Step Solution

Answer

\((r^3 - s^3)^{\frac{1}{3}}\)

Step 1: Identify the Radical Expression

The given expression is \[ \sqrt[3]{r^3 - s^3} \]which is a cube root. Our goal is to rewrite this using rational exponents.

Step 2: Apply the Definition of Rational Exponents

The cube root of any expression \( x \) can be written as \( x^{\frac{1}{3}} \). In the given expression, the expression under the cube root is \( r^3 - s^3 \).

Step 3: Rewrite the Expression Using Rational Exponents

Replace the cube root with its equivalent rational exponent form:\[ (r^3 - s^3)^{\frac{1}{3}} \] This is the rewritten form using rational exponents.

Key Concepts

Cube Root

A cube root is a number that when multiplied by itself three times equals the original number. For example, if you have a number \( x \), the cube root of \( x \) is the number \( y \) such that \( y \, \times \, y \, \times \, y = x \). When we see a cube root symbol, it often looks like this: \( \sqrt[3]{x} \). However, instead of working with this root form, we often find it convenient in algebra to express it using rational exponents. The cube root of an expression can be transformed to a rational exponent by rewriting it as \( x^{\frac{1}{3}} \). This shift from radical to rational exponent makes algebraic manipulations easier in many cases. Recognizing and rewriting cube roots in this form is a key skill in algebra.

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. For example, \( 2^3 \) (read as 'two raised to the power of three') is \( 2 \, \times \, 2 \, \times \, 2 \), which equals 8. When working with exponents, it's important to understand the rules that govern them:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Zero Exponents: Any non-zero number raised to the power of zero is 1, \( a^0 = 1 \).
• Negative Exponents: \( a^{-m} = \frac{1}{a^m} \).

Using these rules, we can simplify expressions and solve equations more easily. Understanding exponents is essential for working with quadratic equations, polynomials, and logarithms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. They represent values and can be simplified or manipulated according to algebraic rules. Basic components of algebraic expressions include terms, coefficients, and variables. Here’s a brief dive into each:

• Terms: Parts of an expression separated by addition or subtraction signs. For example, in \( 3x + 4y - 5 \), there are three terms.
• Coefficients: The numerical factor in a term. In the term \( 3x \), 3 is the coefficient.
• Variables: Symbols that represent unknown quantities, usually letters like \( x \), \( y \), or \( r \).
• Constants: Numbers that stand alone without a variable. Like \( -5 \) in the above example.

Algebraic expressions can be evaluated and simplified following standard mathematical procedures and algebraic properties. This includes factorization, distribution, and combining like terms. Mastery of algebraic expressions provides a foundation for more complex algebraic operations, equation solving, and analysis tasks.