Problem 19 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \left(\frac{1}{13}\right)^{0}\left(\frac{2}{3}\right)^{6}\left(\frac{4}{9}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Evaluate any base to the power of zero

Understand that any non-zero base raised to the power of zero equals one. Therefore, \( \left( \frac{1}{13} \right)^0 \) simplifies to 1. The entire expression now becomes: \( 1 \cdot \left( \frac{2}{3} \right)^6 \cdot \left( \frac{4}{9} \right)^{-3} \).

Step 2: Address negative exponents

Recall that a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. So, \( \left( \frac{4}{9} \right)^{-3} \) becomes \( \left( \frac{9}{4} \right)^3 \). Now the expression is \( 1 \cdot \left( \frac{2}{3} \right)^6 \cdot \left( \frac{9}{4} \right)^3 \).

Step 3: Simplify expressions with positive exponents

Simplify the expression by evaluating each base to the positive exponents: \( \left( \frac{2}{3} \right)^6 = \frac{64}{729} \) and \( \left( \frac{9}{4} \right)^3 = \frac{729}{64} \).

Step 4: Multiply the results

Multiply the simplified expressions: \( 1 \cdot \frac{64}{729} \cdot \frac{729}{64} \). Notice that \( \frac{64}{729} \cdot \frac{729}{64} \) equals 1, because the numerator and the denominator cancel out. So the entire expression evaluates to 1.

Key Concepts

Negative exponents

When you see a negative exponent, remember it's a way to express the inverse of something. Let's say you have a fractional expression like \( \left( \frac{4}{9} \right)^{-3} \). Here, the negative exponent "-3" tells you that you need to take the reciprocal of the fraction first. So, it becomes \( \left( \frac{9}{4} \right)^{3} \).

Why do we need reciprocal?
- Negative exponents indicate that the fraction is flipped upside-down.
- After flipping, you only use the positive exponent to avoid any confusion.
This concept is very useful when you’re simplifying expressions involving powers. By remembering that negative exponents simply denote the reciprocal of a fraction to the positive power, you can easily manage even the toughest problems.

Reciprocal function

The reciprocal function flips a number or fraction over. If you have a fraction like \( \frac{4}{9} \), its reciprocal is \( \frac{9}{4} \). So, when you see a negative exponent, think of this flipping action.

How to find the reciprocal:

• Take the fraction in question.
• Switch the numerator and the denominator.

Once you've flipped the base for negative exponents, you get to deal with positive exponents. If your expression already has a negative base, the reciprocal becomes your best friend.

Multiplying fractions

Now that we've covered flipping and reciprocal, let's dive into multiplying fractions. Consider you have two fractions: \( \frac{a}{b} \) and \( \frac{c}{d} \). When you multiply them, you multiply the numerators together and the denominators together.

For example, multiplying \( \frac{64}{729} \) by \( \frac{729}{64} \):

• Multiply the numerators: \( 64 \times 729 \)
• Multiply the denominators: \( 729 \times 64 \)

Simplifying the process:
- Sometimes, you can cancel numbers from the numerator and the denominator if they are the same, as seen in the expression \( \frac{64}{729} \cdot \frac{729}{64} \) simplifying to 1.
This makes calculations far easier, especially when numbers look overwhelming.

Zero exponent rule

The zero exponent rule is a neat little trick in math. It states that any non-zero base raised to the power of zero equals 1. This might seem surprising at first!

For instance, if you take \( \left( \frac{1}{13} \right)^0 \), it's simply 1. This rule helps simplify expressions quickly.

Why the zero exponent rule works:

• Math is consistent, and the zero exponent rule helps keep it that way.
• Imagine a number being divided by itself. Well, that's precisely what happens with an exponent of zero!

So next time you see a zero exponent, remember: it makes your life easier by reducing the base to just 1.