Problem 20 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \frac{2^{-3}}{3^{0}} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{1}{8}\).

Step 1: Simplify the Denominator

The expression in the denominator is \(3^0\). Any non-zero number raised to the power of 0 is 1. Therefore, \(3^0 = 1\). The expression now becomes \(\frac{2^{-3}}{1}\).

Step 2: Simplify the Numerator

The expression in the numerator is \(2^{-3}\). A negative exponent means that we take the reciprocal, so \(2^{-3} = \frac{1}{2^3}\). Therefore, \(\frac{2^{-3}}{1}\) simplifies to \(\frac{1}{2^3}\).

Step 3: Evaluate the Power of 2

Now calculate \(2^3\). \(2^3 = 2 \times 2 \times 2 = 8\).

Step 4: Final Simplification

Now we have \(\frac{1}{2^3}\), which simplifies to \(\frac{1}{8}\). Thus, the original expression \(\frac{2^{-3}}{3^0}\) evaluates to \(\frac{1}{8}\).

Key Concepts

Understanding Negative Exponents

Negative exponents can be a little confusing at first, but they are actually quite simple! When you see a negative exponent, like in the expression \(2^{-3}\), it just means you're finding the reciprocal, or the "flipped" version, of its positive exponent equivalent. So, \(2^{-3}\) is the same as \(\frac{1}{2^3}\).

• Think of negative exponents as instructions to "look below the line" in a fraction.
• So \(a^{-n}\) becomes \(\frac{1}{a^n}\).

Once you flip the base, you solve for the positive exponent, just as you normally would. This step converts what seems tricky into an easier problem. For instance, finding \(2^3\), which turns into \(2 \times 2 \times 2 = 8\). So, \(2^{-3}\) turns into \(\frac{1}{8}\) when simplified!

Zero Exponent Rule Explained

The zero exponent rule simplifies expressions significantly. It states that any non-zero number raised to the power of zero equals one. This may seem strange at first, but it's incredibly useful and can be seen as a sort of "reset" button for powers.

• For any number \(a\), \(a^0 = 1\) given \(a eq 0\).
• It doesn't matter how big or small the number is; as long as the exponent is zero, the result is always one.

This rule helps simplify expressions quickly, like in the denominator of our example \(3^0\), which becomes 1 with no hesitation. Thus, it streamlines our problems, making them easier to handle.

The Art of Simplifying Expressions

Simplifying expressions involves breaking them down to their most basic form, which makes them easier to work with. The steps often involve applying exponent rules like those for negative exponents and zero exponents. In our expression \(\frac{2^{-3}}{3^0}\), we followed these rules to turn a complicated expression into something neat and easy to understand.

• Simplify each part of the expression separately; break them down one by one.
• Apply exponent rules systematically: address negative exponents, zero exponents, and any multiplication or division.

Once simplified, expressions aren't just more manageable, they also convey meaningful information much more clearly. It’s all about making it simple enough to work with, yet powerful enough to give results swiftly.

Calculating the Power of a Number

Calculating the power of a number is a fundamental math skill that often supports more complex mathematical ideas like exponents and algebra. The power of a number asks you to multiply that number by itself a number of times indicated by the exponent.

• For example, \(2^3\) means multiplying 2 by itself three times: \(2 \times 2 \times 2\).
• It focuses on repeated multiplication to represent large numbers easily.

When you understand how to calculate the power of a number, working with expressions becomes more straightforward. In our solution, finding \(2^3 = 8\) allowed us to turn \(\frac{1}{2^3}\) into \(\frac{1}{8}\), the final, simplified value of the original expression.