Problem 12 - Algebra 2

Question

Simplify each expression. $$ \frac{a^{2}}{b^{\frac{1}{3}}} \cdot \frac{b}{a^{\frac{1}{2}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(a^{\frac{3}{2}} b^{\frac{2}{3}}\).

Step 1: Write the Expression

Let's start by rewriting the given expression:\[\frac{a^{2}}{b^{\frac{1}{3}}} \cdot \frac{b}{a^{\frac{1}{2}}}\]

Step 2: Combine the Fractions

Combine the fractions by multiplying the numerators together and the denominators together:\[\frac{a^{2} \cdot b}{b^{\frac{1}{3}} \cdot a^{\frac{1}{2}}}\]

Step 3: Simplify the Expression by Exponent Properties

Apply the properties of exponents. For the numerator and denominator:Numerator: \(a^2 \cdot b = a^2 b\)Denominator: \(b^{\frac{1}{3}} \cdot a^{\frac{1}{2}} = a^{\frac{1}{2}} b^{\frac{1}{3}}\) So the expression simplifies to:\[\frac{a^{2}b}{a^{\frac{1}{2}}b^{\frac{1}{3}}}\]

Step 4: Simplify Using Exponent Laws

We simplify using the rule \(\frac{x^m}{x^n} = x^{m-n}\):- Simplify for \(a\):\(a^{2-\frac{1}{2}} = a^{\frac{3}{2}}\)- Simplify for \(b\):\(b^{1-\frac{1}{3}} = b^{\frac{2}{3}}\)So the final simplified expression is:\[a^{\frac{3}{2}} b^{\frac{2}{3}}\]

Key Concepts

Exponent Properties

Exponent properties help us handle expressions involving repetitive multiplication more easily. In our expression, we work with terms like \(a^2\), \(b^{\frac{1}{3}}\), and \(a^{\frac{1}{2}}\). Each exponent tells us how many times to use the base in a multiplication.

The key property we use in exponentiation is that when you multiply like bases, you add their exponents together. For example, \(a^m \cdot a^n = a^{m+n}\). Similarly, when you divide like bases, you subtract the exponents: \(\frac{x^m}{x^n} = x^{m-n}\). Consider our example:

• The term \(a^2\) means \(a\) is multiplied by itself, two times.
• The term \(a^{\frac{1}{2}}\) represents the square root of \(a\) (since raising to the power of \(\frac{1}{2}\) is the same as taking a square root).
• For the \(b\) terms, \(b^{\frac{1}{3}}\) can be considered as the cube root of \(b\).

Applying these properties correctly allows us to rewrite and eventually simplify expressions.

Fraction Simplification

Fraction simplification is the process of making a fraction easier to interpret. In mathematics, we often want expressions in their simplest form. This involves canceling common factors out from the numerator and the denominator.

In the given expression \[\frac{a^{2} \cdot b}{b^{\frac{1}{3}} \cdot a^{\frac{1}{2}}}\]our main goal was to break it into manageable parts by factoring and canceling. By multiplying fractions, we consolidate numerators and denominators:

• Multiply the top numbers (numerators) together.
• Multiply the bottom numbers (denominators) together.
• Simplify by canceling common terms from numerator and denominator.

For example, \(b\) in the numerator and denominator can be simplified using exponent properties (which we apply next). This makes complex fractions easier to handle and less intimidating.

Laws of Exponents

The laws of exponents guide us through the manipulation of powers and roots in algebra. These rules make working with exponential expressions straightforward and efficient.

In our exercise, after simplifying the fractions, we employed the laws of exponents to clean up the result even further:

• When dividing two terms with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• When dealing with roots, remember that roots are just fractional exponents: the square root is \(\frac{1}{2}\) and the cube root is \(\frac{1}{3}\).

Let's see how these work in simplifying the expression:

• For \(a\), we have \(a^{2-\frac{1}{2}} = a^{\frac{3}{2}}\).
• For \(b\), \(b^{1-\frac{1}{3}} = b^{\frac{2}{3}}\).

This ability to transform a complex expression into a simpler one relies heavily on the mastery of exponent laws, enabling us to express the final solution as \(a^{\frac{3}{2}} b^{\frac{2}{3}}\). Mastering these laws elevates your algebra skills, making math problems much more manageable.