Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ a^{4 / 3} a^{-3 / 2} a^{1 / 6} $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Understanding Power Rules

To simplify expressions involving exponents, you need to remember the power rule: \( a^m \times a^n = a^{m+n} \). This means when you multiply powers with the same base, you add their exponents.

Step 2: Adding the Exponents

Given the expression \( a^{4/3} \times a^{-3/2} \times a^{1/6} \), apply the power rule by adding the exponents: \( \frac{4}{3} + \left( -\frac{3}{2} \right) + \frac{1}{6} \).

Step 3: Simplifying the Addition of Fractions

To add these fractions, we need a common denominator. The denominators are 3, 2, and 6. The least common denominator is 6. Convert each fraction:- \( \frac{4}{3} = \frac{8}{6} \) - \( -\frac{3}{2} = -\frac{9}{6} \) - \( \frac{1}{6} = \frac{1}{6} \).Now add them: \( \frac{8}{6} + (-\frac{9}{6}) + \frac{1}{6} = \frac{8 - 9 + 1}{6} = 0 \).

Step 4: Final Expression with Zero Exponent

Since the sum of the exponents is 0, by the rule \( a^0 = 1 \) (for \( a eq 0 \)), the expression \( a^0 = 1 \).

Key Concepts

Power Rule

The Power Rule is a key concept in simplifying exponential expressions. It states that when you multiply two expressions with the same base, you simply add their exponents.
This rule can be summarized as:

• \( a^m \times a^n = a^{m+n} \)

This rule is incredibly useful because it helps reduce complex expressions to simpler forms.
In the original exercise, applying the Power Rule allows us to combine the three separate powers of \( a \) into a single power.
Remember, this rule only works when the bases are identical. This consistency with the base is crucial for the Power Rule to hold.

Fraction Addition

Adding fractions can be tricky at first, but it's much easier once you find a common denominator.
This denominator should be a number that all the individual denominators can divide evenly.
For example, if you're adding \( \frac{4}{3} \), \( -\frac{3}{2} \), and \( \frac{1}{6} \), the least common denominator is 6.
To convert each fraction:

• \( \frac{4}{3} \) becomes \( \frac{8}{6} \)
• \( -\frac{3}{2} \) becomes \( -\frac{9}{6} \)
• \( \frac{1}{6} \) stays \( \frac{1}{6} \)

After aligning the denominators, you add the numerators.
This process results in: \[ \frac{8}{6} + (-\frac{9}{6}) + \frac{1}{6} = \frac{8 - 9 + 1}{6} = 0 \]Finding a common denominator makes fraction addition straightforward and simplifies the process of handling fractional exponents.

Zero Exponent Rule

The Zero Exponent Rule is a handy shortcut when working with powers.
This rule states that any non-zero number raised to the power of zero equals one.
Mathematically, it's shown as:

• \( a^0 = 1 \)

This principle comes from the division of exponents with equal bases:

• \( a^n / a^n = a^{n-n} = a^0 = 1 \)

So the rule isn't just convenient; it has a solid theoretical background.
In our exercise, we ended up with \( a^0 \) after simplifying the exponent sum, resulting in the expression simplified to 1.
The Zero Exponent Rule is a powerful tool that often appears in algebra, easy to remember and often simplifies expressions quickly.