Problem 87 - Algebra 2

Question

Simplify. Assume that no variable equals zero. \(\left(\frac{b^{7}}{a^{4}}\right)^{0}\)

Step-by-Step Solution

Answer

1

Step 1: Understand the expression

The expression given is \(\left(\frac{b^{7}}{a^{4}}\right)^{0}\). We need to simplify this expression.

Step 2: Apply the Zero Exponent Rule

According to the zero exponent rule, any non-zero base raised to the power of zero equals 1. Thus, \(\left(\frac{b^{7}}{a^{4}}\right)^{0} = 1\). This holds true because no variable equals zero as per the assumption given in the problem.

Step 3: Simplify the expression

Using the zero exponent rule, the expression simplifies to 1, as any non-zero term raised to the power of zero is 1.

Key Concepts

Exponent Rules

Understanding exponent rules is essential for simplifying mathematical expressions. In mathematics, an exponent tells us how many times a base is used as a factor in a multiplication. The exponent rules help us manipulate these expressions easily.
One of the key rules is the zero exponent rule, which states that any non-zero number raised to the power of zero equals one. This might seem surprising at first, but it simplifies many expressions in algebra.

• Example: \(a^0 = 1\) for any \(a eq 0\).
• Example: \((rac{b^{7}}{a^{4}})^{0} = 1\).

Using these rules, especially the zero exponent rule, helps us recognize patterns and simplify expressions efficiently.

Expression Simplification

Simplifying expressions is all about making them easier to work with or understand. The goal is to express the problem in its simplest form. When you simplify an expression with exponents, you often apply exponent rules. The zero exponent rule is a great tool for this.
To simplify the expression \((\frac{b^{7}}{a^{4}})^{0}\), follow these steps:

• Identify the expression and the exponent, which is zero in this case.
• Recall that any non-zero number to the power of zero is 1.
• Apply the rule to simplify the entire expression to 1.

Understanding why these rules work can strengthen your algebra skills, making it easier to simplify even more complex expressions.

Mathematical Assumptions

In algebra, assumptions are crucial when simplifying expressions. They tell us what conditions need to be true for the rules we use to work. For instance, when applying the zero exponent rule, the requirement is that the base is non-zero.
In the problem \((\frac{b^{7}}{a^{4}})^{0}\), assuming no variable equals zero ensures that each base, both \(b^7\) and \(a^4\) in the fraction, is non-zero. This validates our use of the zero exponent rule.

• Assumptions help avoid mathematical errors that could arise from undefined expressions like division by zero.
• Understanding these assumptions is key for correctly applying exponent rules and simplifying expressions.

Next time you simplify an expression, consider all assumptions carefully to ensure accuracy.