Problem 54 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ (-y)^{0} n $$

Step-by-Step Solution

Answer

The simplified expression with positive exponents is \(n\).

Step 1: Apply Zero Exponent Rule

Apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is 1. Therefore, \((-y)^{0}\) becomes 1. The expression now becomes \(1 × n\).

Step 2: Simplify the Expression

Simplify \(1 × n\) to \(n\), because multiplying any number by 1 leaves the number unchanged.

Key Concepts

Zero Exponent Rule

One of the foundational concepts in dealing with exponents is the zero exponent rule. This rule states that any non-zero number raised to the power of zero equals one. This might seem a bit counterintuitive at first but think of it as a part of the pattern with exponents. With positive powers, you are essentially multiplying the number by itself a number of times. As you progress down towards zero,
you are effectively dividing out the number by itself with each step, and what's left is one.

• If you have an expression like \(a^0\), it simplifies to \(1\), given that \(a\) is not zero.
• For example, \(5^0\) equals 1, and so does \((-7)^0\).

Expression Simplification

Simplifying expressions is an important skill in algebra. It's all about reducing an expression to its simplest form without changing its fundamental value or meaning. After applying the zero exponent rule, many expressions become centers on multiplication by one, which makes simplification straightforward.
Consider our example, after turning \((-y)^0\) into one,
we were left with the operation \(1 \times n\). The rule here is simple:

• Any number multiplied by one remains the same.
• Thus, \(1 \times n = n\).

To ensure proper simplification:

• Always perform operations step by step to avoid mistakes.
• Check that each part of the expression follows the rules of arithmetic.

Exponents

Exponents are a way to express repeated multiplication of the same number. They consist of a base and an exponent. When you see \(x^y\),
x is the base, and y is the exponent, indicating x is multiplied by itself y times.
They can drastically simplify long multiplication problems.

• If you have \(2^3\), it translates to \(2 \times 2 \times 2 = 8\).
• When an exponent is zero, the result simplifies based on the zero exponent rule.

Understanding exponents also involves knowing how to manipulate them with different rules:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}, a eq 0\)
• Power Rule: \( (a^m)^n = a^{m \cdot n}\)

Mastering these rules will make working with exponents much simpler.