Problem 77 - Algebra 2 and Trigonometry

Question

Find the value of \((-5 a)^{0}-5 a^{-2}\) if \(a=3\)

Step-by-Step Solution

Answer

The value of the expression is \(\frac{4}{9}\).

Step 1: Substitute the Value of a

First, substitute the given value of \(a=3\) into the expression: \((-5 \times 3)^0 - 5 \times 3^{-2}\).

Step 2: Evaluate the Zero Exponent

Recall that any number raised to the power of 0 is 1. Therefore, \((-5 \times 3)^0 = 1\).

Step 3: Evaluate the Negative Exponent

Apply the rule \(x^{-n} = \frac{1}{x^n}\) to \(3^{-2}\):\[5 \times 3^{-2} = 5 \times \frac{1}{3^2} = 5 \times \frac{1}{9} = \frac{5}{9}\].

Step 4: Combine Results

Substitute the evaluated terms back into the expression: \(1 - \frac{5}{9}\). To subtract these, convert 1 to a fraction with a denominator of 9: \(\frac{9}{9}\).

Step 5: Perform the Subtraction

Subtract the fractions: \(\frac{9}{9} - \frac{5}{9} = \frac{4}{9}\). So the value of the expression is \(\frac{4}{9}\).

Key Concepts

Zero Exponent Rule

The zero exponent rule is one of the fundamental principles in algebra. It states that any non-zero number raised to the power of zero is always equal to one. This concept might seem puzzling at first, but it makes sense when you think about the pattern of decreasing exponents: as you reduce the exponent of a base down to zero, you are essentially dividing by the base repeatedly, which consolidates to 1. Consider a few examples to illustrate this point:

• \( 2^3 = 8 \)
• \( 2^2 = 4 \)
• \( 2^1 = 2 \)
• \( 2^0 = 1 \)

This rule is consistent and applies universally, as demonstrated in the exercise when evaluating \((-5 \times 3)^0\), resulting in 1. Understanding this concept is crucial in simplifying expressions with zero exponents effortlessly.

Negative Exponents

Negative exponents can seem tricky at first, but they become simple once you grasp the underlying rule: a negative exponent indicates that you take the reciprocal of the base raised to the positive equivalent of that exponent. Essentially, any base with a negative exponent can be transformed to its positive form by flipping it into a fraction. Consider the rule:

• \( x^{-n} = \frac{1}{x^n} \)

Let's see how this works in the exercise with \(5 \times 3^{-2}\):

• Convert \(3^{-2}\) using the negative exponent rule: \(\frac{1}{3^2}\)
• Compute \(3^2\) as 9
• Multiply the result by 5: \(5 \times \frac{1}{9} = \frac{5}{9}\)

Once you understand this flipping action, simplifying expressions with negative exponents becomes much more manageable.

Substitution Method

The substitution method is a way to simplify algebraic expressions by replacing variables with their corresponding values. It allows for precise calculation and clarity in solving expressions or equations. When you are given a specific value for a variable, you substitute it in to evaluate the expression. In the exercise, we used this method:

• Given \(a = 3\), the initial expression \((-5a)^0 - 5a^{-2}\) requires substitution.
• Replace \(a\) with 3, transforming it into \((-5 \times 3)^0 - 5 \times 3^{-2}\).

This method is handy in breaking down problems into smaller, more manageable steps, making complex equations into straight-forward arithmetic.

Fraction Operations

Fraction operations often appear intimidating, but they become clearer with understanding and practice. Fundamental fraction operations include addition, subtraction, multiplication, and division. Each operation requires a certain level of familiarity with common denominators and equivalent fractions. In the context of the exercise, consider subtracting fractions:

• When you have fractions with the same denominator, like \(\frac{9}{9}\) and \(\frac{5}{9}\), the subtraction is straightforward: \(\frac{9}{9} - \frac{5}{9} = \frac{4}{9}\).
• If they had different denominators, you'd first find a common denominator before performing subtraction.

Remember, fractions represent division. Thus, they are another way to express numbers and to perform operations directly corresponding to our arithmetic rules. With these basics, navigating through fraction operations becomes a less daunting task.