Problem 48 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(5 \sqrt{a b}\)

Step-by-Step Solution

Answer

5 times \(a^{\frac{1}{2}} b^{\frac{1}{2}}\).

Step 1: Identify the Expression Inside the Square Root

The given expression is \(5\sqrt{ab}\). The part inside the square root is \(ab\), which needs to be expressed with a positive rational exponent.

Step 2: Convert the Square Root to a Positive Rational Exponent

Recall that the square root function can be written as a rational exponent. Specifically, \(\sqrt{x} = x^{\frac{1}{2}}\). For \(ab\), this becomes \((ab)^{\frac{1}{2}}\).

Step 3: Distribute the Exponent over the Product

Using the property \((xy)^n = x^n y^n\), apply the exponent to both \(a\) and \(b\). This gives \((ab)^{\frac{1}{2}} = a^{\frac{1}{2}} b^{\frac{1}{2}}\).

Step 4: Include the Constant Factor

The expression originally included a constant factor of 5. So the complete expression becomes \(5 \times a^{\frac{1}{2}} b^{\frac{1}{2}}\).

Key Concepts

Rational Exponents

Rational exponents provide an alternate way to express roots such as square roots. Instead of writing a square root symbol \( \sqrt{} \), we can use an exponent of \( \frac{1}{2} \). This expresses the same mathematical operation but uses the rules of exponentiation.
For example, the square root of any number \( x \) can be written as \( x^{\frac{1}{2}} \). This approach not only simplifies the notation but also makes it easier to perform algebraic manipulations involving roots.

• A rational exponent like \( x^{\frac{a}{b}} \) means that the base \( x \) is raised to the power \( a \) and then the \( b\)-th root is taken.
• It can also be read in two steps: power and root. For instance, \( x^{\frac{1}{3}} \) means the cube root of \( x \).
• This method integrates well with rules for multiplying and dividing exponents, which can simplify complex expressions.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It's commonly written with the radical symbol \( \sqrt{} \). However, expressing square roots with rational exponents can simplify understanding and calculating these roots, especially in algebraic contexts.
For instance, \( \sqrt{ab} \) becomes \( (ab)^{\frac{1}{2}} \) when written with a rational exponent. This allows us to use the property of distributing exponents over products. It's a key principle for breaking down expressions into simpler, more manageable parts.

• Mathematically, it helps to see the square root operation as raising a number to the \( \frac{1}{2} \) power.
• Instead of dealing with the square root function directly, it provides a consistent way of handling roots of all degrees through exponentiation.
• By using this exponent form, complex expressions involving roots can be worked out using familiar exponent rules.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and the operations that connect them. When working with such expressions, converting roots and other complex components into simpler forms using rational exponents can be particularly helpful.
An expression like \( 5 \sqrt{ab} \) becomes more manageable when written as \( 5 \times a^{\frac{1}{2}} b^{\frac{1}{2}} \). This reformulation is critical in algebra because it allows us to apply the exponent rules more easily, opening up possibilities for simplification and manipulation.

• The new form using rational exponents renders it easier to distribute exponents across variables.
• Additionally, operations such as multiplication and division become more straightforward, thanks to consistent exponent routines.
• These transformed expressions can also facilitate solving complex equations by simplifying how we deal with roots across different terms.