Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ 7^{-2} $$

Step-by-Step Solution

Answer

The number \(7^{-2}\) rewritten without using exponents is \(\frac{1}{49}\).

Step 1: Evaluate the Negative Exponent

To rewrite the number without using exponents, we need to remember that any base to a negative exponent is equal to the reciprocal of the base raised to the same positive exponent. So, for a base \(a\) having the exponent \(-n\), the formula is: $$ a^{-n} = \frac{1}{a^n} $$ Using this rule, we can rewrite the given number with a positive exponent.

Step 2: Substitute in the Given Number

Now, let's use the formula to rewrite \(7^{-2}\). In our case, the base \(a = 7\), and the exponent \(-n = -2\). According to the formula: $$ 7^{-2} = \frac{1}{7^2} $$

Step 3: Calculate the Denominator using the Positive Exponent

Now, we will calculate the denominator of the fraction by raising the base 7 to the positive exponent 2: $$ 7^2 = 7 \times 7 = 49 $$

Step 4: Rewrite the Number without Using Exponents

After we have calculated the denominator, we can replace it in the fraction. So, the final expression without using any exponents is: $$ \frac{1}{7^2} = \frac{1}{49} $$ So, the number \(7^{-2}\) rewritten without using exponents is \(\frac{1}{49}\).

Key Concepts

Exponentiation Rules

Exponentiation is a fundamental concept in mathematics used to describe repeated multiplication. For example, the expression \(a^n\) means that the base \(a\) is multiplied by itself \(n\) times. However, negative exponents introduce an additional rule.
If the exponent is negative, it signifies the reciprocal of the base with a positive exponent. This is captured in the formula \(a^{-n} = \frac{1}{a^n}\).
Let's remember these basic rules:

• When multiplying like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• A base with a zero exponent is always 1: \(a^0 = 1\), where \(a eq 0\).

Understanding these rules can help simplify expressions and solve complex mathematical problems with ease.

Reciprocal of a Number

The reciprocal of a number is essentially its multiplicative inverse. This means that when a number is multiplied by its reciprocal, the result is always 1. For example, the reciprocal of 5 is \(\frac{1}{5}\), and multiplying them yields \(5 \times \frac{1}{5} = 1\).
For expressions with negative exponents, finding the reciprocal comes naturally. A negative exponent means you take the reciprocal of the base and then apply the positive of that exponent. For instance, with \(7^{-2}\), the reciprocal is \(\frac{1}{7}\), and raised to the power of 2 gives \(\frac{1}{7^2}\).
This reciprocal concept is foundational for understanding how negative exponents work, ensuring that expressions can be rewritten in a form that is often more straightforward to compute.

Simplifying Exponents

Simplifying exponents can make mathematical expressions easier to handle and understand. Let's break down why it's important.
When an expression like \(7^{-2}\) is simplified, it converts to a more basic arithmetic form: \(\frac{1}{49}\). This simpler form is easier to interpret and use in further calculations.
The simplification process typically involves the following steps:

• Apply the reciprocal rule for negative exponents to transform them into a positive exponent fraction.
• Calculate the power of the positive exponent, such as \(7^2\) to get 49.
• Express the initial problem, now free of any exponentiation, stating \(\frac{1}{49}\).

This flow enhances comprehension and facilitates computations, especially when dealing with larger mathematical expressions or equations.