Problem 14 - Intermediate Algebra

Question

Simplify each numerical expression. \(-\left(\frac{5}{6}\right)^{0}\)

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Understanding Zero Exponents

Any non-zero number raised to the power of zero is always equal to 1. Therefore, the expression \(rac{5}{6}^{0}\) simplifies directly to 1.

Step 2: Apply the Negative Sign

The original expression is \(-\left(\frac{5}{6}\right)^{0}\). Once we have simplified \(\left(\frac{5}{6}\right)^{0}\) to 1, we must consider the negative sign outside the parentheses. Thus, \(-1\) is the simplified form of \(-\left(\frac{5}{6}\right)^{0}\).

Key Concepts

Zero Exponent Rule

The Zero Exponent Rule is a fundamental principle in algebra that states any non-zero number raised to the power of zero equals one. This might seem puzzling at first, but it can be understood by looking at the patterns in exponents. For instance:

• When we raise a number to a descending sequence of exponents like 3, 2, then 1, we divide by the base each time. For example: \[2^3 = 8, \ 2^2 = 4, \ 2^1 = 2\]Notice the pattern? Each step divides the result by 2.


• Extending that pattern, to reach an exponent of zero, we divide one more time by the base: \[2^0 = \frac{2^1}{2} = 1\]So, \[\left(\frac{5}{6}\right)^{0} = 1\]no matter the value of the base, as long as it's not zero.

This rule makes calculations a lot simpler since any complex expression with an overall exponent of zero quickly resolves to 1.

Negative Sign in Expressions

Negative signs in mathematical expressions change the direction or sign of the result. When you see an expression like \[-\left(\frac{5}{6}\right)^{0}\], you need to pay attention to the placement of the negative sign.
Here, the negative sign is outside the parentheses, and it doesn't affect the initial simplification of \[\left(\frac{5}{6}\right)^{0}\]. You first apply the zero exponent rule:

• The expression inside the parentheses simplifies to 1 due to the zero exponent rule.
• Only after simplifying the inner expression do we apply the negative sign to the result.

This results in:\[-1\]By following these steps, you ensure that negative signs are correctly applied, reflecting the true intent behind the expression.

Numerical Simplification

Numerical simplification involves reducing expressions to their simplest form to make calculations easier and clearer. This process often includes applying known mathematical rules such as exponents or dealing with negative signs.

• First, identify any rules that apply, such as the zero exponent rule in our exercise.


• Perform operations step by step to simplify the expression bit by bit.


• If there are additional operations like multiplication or a negative sign, make sure these are applied last to the already simplified components.

In our exercise, we followed the rule for zero exponents to simplify the fraction \[\left(\frac{5}{6}\right)^{0}\] into 1, then applied the negative sign to arrive at the final simplified answer of \[-1\]. Simplifying expressions like this not only makes them easier to work with but also helps in understanding the effects of mathematical operations on numbers.