Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ (-0.3)^{-2} $$

Step-by-Step Solution

Answer

\( (-0.3)^{-2} = \frac{1}{(-0.3)^2} = \frac{1}{0.09} = 11.\overline{1} \)

Step 1: Rewrite the expression using the negative exponent rule

Because the given exponent is negative, we rewrite the expression using the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^{n}}\). In our case, the base is -0.3 and the exponent is -2. \[ (-0.3)^{-2} = \frac{1}{(-0.3)^2} \]

Step 2: Simplify the expression

Now, we simplify the expression by raising the base (-0.3) to the 2nd power: \[ \frac{1}{(-0.3)^2} = \frac{1}{(-0.3)(-0.3)} \] Since -0.3 multiplied by -0.3 results in a positive value, we can find the product and simplify: \[ \frac{1}{(0.09)} \]

Step 3: Find the reciprocal of the denominator

To rewrite the number without using exponents, we find the reciprocal of the denominator, which is the same as dividing 1 by the denominator (0.09): \[ \frac{1}{0.09} = 11.\overline{1} \] So, the number \((-0.3)^{-2}\) rewritten without using exponents is 11.\overline{1}.

Key Concepts

Negative Exponent Rule

When dealing with negative exponents, it’s important to understand a simple but crucial rule: the negative exponent rule. This rule essentially tells us how to transform an expression that has a negative exponent into a more workable form. The rule states that any number raised to a negative exponent can be rewritten as the reciprocal of that number raised to the corresponding positive exponent.
For example, if you have an expression like \((-0.3)^{-2}\), the negative exponent rule instructs us to write it as \(\frac{1}{(-0.3)^2}\).
In essence:

• \(a^{-n} = \frac{1}{a^{n}}\)

This switch from the negative to the positive exponent makes it easier to perform calculations and simplifies the process significantly.
By applying this rule, we effectively eliminate the negative exponent and bring the problem into a more familiar form that’s typically easier to work with.

Simplifying Expressions

Simplifying expressions is a fundamental process in algebra and mathematics in general. Once we have transformed an expression using the negative exponent rule, the next step is to simplify it. This involves performing the arithmetic operations necessary to reduce the expression to its simplest form.
Continuing with our example, after applying the negative exponent rule, we have:\(\frac{1}{(-0.3)^2}\).
Here, we need to calculate \((-0.3)^2\), which involves multiplying \(-0.3\) by itself.
Crucially, note that multiplying two negative numbers together gives a positive result. So:

• \(-0.3 \times -0.3 = 0.09\)

After this calculation, the expression simplifies to \(\frac{1}{0.09}\).
Simplifying expressions helps in streamlining the problem-solving process and makes it easier to see what the final result will be.

Reciprocal Calculation

Understanding and calculating reciprocals is another core concept that ties directly into working with negative exponents and simplifying expressions. A reciprocal of a number is basically 1 divided by that number.
In our context, finding the reciprocal allows us to invert fractions, which is crucial when dealing with expressions like \(\frac{1}{0.09}\).
Calculating the reciprocal of \(0.09\) means finding a number that, when multiplied by \(0.09\), results in \(1\). This is effectively what division is performing in this scenario:

• \(\frac{1}{0.09} = 11.\overline{1}\)

By performing the reciprocal calculation, you can transform the expression into its simplest numerical form.
Understanding this concept is useful not only for expressions with negative exponents but also for a wide range of mathematical problems involving fractions and division.